THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR THE SYNTHESIS OF NONUNIFORM TRANTSMIISSION J'i.ES Technical'Report No. 116 2899-,41-T Cooley Electronics Laboratory Department of Electrical Engineering By Pieter G. Cath Approved by: Charles B. Sharpe Project 2899 TASK ORDER NO. EDG-4 CONTRACT NO. DA-36-039 sc-78283 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3A99-06-ool001-01l Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan January 1961

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ACKNOWLEDGMENTS The author is deeply indebted to the Chairman of his Doctoral Committee, Professor C. B. Sharpe, who not only aroused his interest in the problem of nonuniform lines, but who also was a source of constant encouragement and of valuable suggestions during the course of the research. The author also wishes to thank the other members of his Doctoral Committee for their helpful comments concerning the material. Finally, the author would like to thank Miss A. M. Rentschler, Mrs. A. M. Brockus, Mr. R. E. Graham, and Mr. E. J. DeRienzo who prepared the manuscript for publication. A part of the research reported in this dissertation was supported by the U. S. Army Signal Corps. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS i i LIST OF TABLES v LIST OF FIGURES v LIST OF SYMBO3LS vii ABSTRACT xi CHAPTER I. INTRODUCTION 1 CHAPTER II. STATEMENT OF THE PROBLEM 3 CHAPTER III. REVIEW OF THE LITERATURE 6 3.1 Introduction 6 3.2 The Differential Equations for Nonuniform Lines 7 3.2.1 The Approximate Solution for Terminated Lines 9 3.2.2 The Approximate Solution in the Presence of Load Reflections 12 3.3 Applications of the Fourier Transform 14 3.3.1 The Synthesis of Impedance Transformers 14 3.3.2 Nonuniform Lines as Filters 19 CHAPTER IV. THE SYNTHESIS OF MATCHING SECTIONS 22 4.1 Introduction 22 4.2 The General Synthesis Problem 23 4.2.1 The Case of a Real Generator Impedance 26 4.2.2 Synthesis of Driving Point Impedances 27 4.3 The Determination of the Reflection-Distribution Function 28 4.3.1 Evaluation of F [cos nay] 29 4.3.2 Evaluation of F [sin nry] 33 4.3.3 Impedance Transformation 37 4.3.4 Summary of Results 38 4.4 The Determination of the Characteristic-Impedance Function 42 CHAPTER V. THE THEORY OF DISCRETE CHEBYSHEV APPROXIMATION 44 5.1 Introduction 44 5.2 Reduction to an Overdetermined System of Linear Equations 44 5.3 Theory of Overdetermined Systems of Linear Equations 46 5.3.1 Determination of the Center of a Reference 50 5.3.2 Overdetermined Systems with Constraints 51 5.3.3 The Replacement Procedure 53 5.3.4 Example 58 11i

TABLE OF CONTENTS (continued) CHAPTER VI. EXAMPLES 62 6.1 Introduction 62 6.2.1 0.75 X Transformer 65 6.2.2 1 X Transformer 72 6.3 Synthesis of a Matching Section 80 CHAPTER VII. CONCLUSIONS 100 APPENDIX A METHOD FOR IMPROVED CONVERGENCE 102 LIST OF REFERENCES 107 DISTRIBUTION LIST log09 iv

LIST OF TABLES Page TABLE I The functions C (s) and Cn (s) for integral values of (4s). 63 TABLE II The functions S e (s) and SO (s) for integral values of (4s). 64 LIST OF FIGURES Figure Page 2.1 Circuit configuration for the matching problem. 3 3.1 Reflection pattern for p(y) = 1 - 0.636 cos 2ry. 17 3.2 Reflection pattern for p(y) = 1 - 0.889 cos 2ry + 0.0112 cos 4~cy. 17 3.3 Theoretical and experimental filter behavior for P(Y) = 3.3 cos 2ity. 21 3.4 Theoretical and experimental filter behavior for p(y) = 6.6 cos 4iy. 21 4.1 Circuit configuration for the matching problem. 24 4.2 Plot of the function C4(s). 32 4.3 Plot of the function S5(s). 36 6.1 Reflection pattern for a 0.75 X transformer. 70 6.2 Reflection-distribution function for a 0.75 X transformer. 71 6.3 Characteristic-impedance function for a 0.75 x transformer. 71 6.4 Reflection pattern for a 1 X transformer. 80 6.5 Reflection-distribution function for a 1 x transformer. 81 6.6 Characteristic-impedance function for a 1 X transformer. 81 6.7 Circuit diagram for the matching example. 82 6.8 Reflection-distribution function for matching section No. 1. 91 6.9 Characteristic-impedance function for matching section No. 1. 91 v

LIST OF FIGURES (Continued) Figure Page 6.10 Cross section of matching section No. 1. 92 6.11 Plot of the function ReIlej2v} and its approximation. 96 6.12 Plot of the function Im{rleJ2 s} and its approximation. 96 6.13 Reflection-distribution function for matching section No. 2. 97 6.14 Characteristic-impedance function for matching section No. 2. 97 6.15 Cross section of matching section No. 2. 98 vi

LIST OF SYMBOLS Symbol Description Defined or first used in equation Cn(s) Function generated by the Fourier transform of cos nry (4.17) Cne(S) Cne(s) = Cn(s) for even n (4.17) Cno(s) Cno (s) = C (s) for odd n (4.18) E. Equation replacing one equation (Er) of the old 1 reference (5r33) Ek Equation denoting the kth plane in n-dimensional space (1 < k < m) (5.7) Eo Denotes constraint equation (5.21) Er Plane of the old reference to be replaced by Ei (535) F(s) Function to be approximated (5.2) F[p] Complex Fourier transform of P (4.1) G(s) Fourier transform of P(y) (4.13) I(x) Current in transmission line at the point x (3.1) R1 Real part of Z1 (4.10) S (s) Function generated by the Fourier transform of sin niy (4.28) Sne (s) Sne(s) = Sn(s) for even n (4.28) Sn(s) Sn(s) = Sn(s) for odd n (4.29) U(x) Voltage in transmission line at the point x (3.2) Un(s) Transformation of C (s) (A.2) Vn(s) Transformation of S (s) (A.12) Z0(y) Characteristic-impedance function (3.5) Zo1 ZO1 = ZO(O); characteristic impedance at input of the nonuniform line (4.5) Z02 Z02 = ZO(Y=l); characteristic impedance at receiving end of the nonuniform line (3.23) vii

LIST OF SYMBOLS (Continued) Symbol Description Defined or first used in equation 1 Z1 = Z1(s ); internal impedance of generator (4.5) Z2 Z2 = Z2(s); load impedance (4.6) akj Direction numbers of the plane Ek (1 < j < n) (5.5) a Coefficients of the cosine terms in expansion (4.12) (3.31) ane ane = a for even n (4.21) ane ne n a a = a for odd n (4.21) no no n b Coefficients of the sine terms in expansion n (4.12) (3.27) b b = b for even n (4.32) ne ne n b b = b for odd n (4.32) no no n Ck Constant term in equation Ek (5.6) di Diameter of center conductor (4.50) do Inside diameter of outer conductor (4.50) f(s) Function approximating F(s) (5.1) fj(s) Approximating functions (1 < j < n) (5.1) h(s) Approximation error (5.2) h Reference error (5.18) h' Reference error of the new reference (5.30) hk Error, or residue, of equation Ek (5.8) kh Residues belonging to the new reference (5.37) X Length of nonuniform transmission line (3.20) ne Stands for "only even values of n" (4.17) no Stands for "only odd values of n" (4.18) nk Vector normal to the plane Ek (5.9) viii

LIST OF SYMBOLS (Continued) Symbol Description Defined or first used in equation s Frequency variable; s = - (3.21) Sk kth sampling point of the variable s (5.3) u Transformation of a (A.2) n n v Transformation of b (A.12) n n x Independent variable of length along the transmission line (3.1) x. Coefficients in the expansion (5.1) corresponding to the jth coordinate in n-dimensional space (5.1) y Normalized variable of length; y =? (3.20) y(x) Shunt admittance per unit length at the point x (3.1) z(x) Series impedance per unit length at the point x (3.2) r(x) Reflection-coefficient function (3.7) p r = r (s); reflection coefficient at the input of the nonuniform transmission line (3.17) rI Load reflection referred back to input of line (3.24) r1(s) Function of F(s) and r (s) (4.4) 2 r2 = P2(s); reflection coefficient at the load (3.23),(x) Imaginary part of y(x) (3.18) 7(x) Propagation-constant function (3.6) x Wavelength of electromagnetic wave (3.21) kk Non-zero constants (1 < k < n+l) (5.10) Constants belonging to the new reference (5.35) Constant coefficients (5.33) p(y) Reflection-distribution function (3.13) ix

ABSTRACT A general synthesis procedure has been developed by which matching sections can be synthesized. A matching section is a section of lossless nonuniform transmission line of finite length that provides a match between a generator with complex internal impedance and a complex load impedance, such that maximum power transfer is achieved over a specified range of frequencies. The values of the internal impedance of the generator and the load impedance can be given either in equation form or in the form of measurements. Special cases that can be treated with the general method include impedance transformers, for which generator and load impedance are both real, and driving point impedances that must exhibit a certain behavior over a specified range of frequencies. Existing methods to synthesize impedance transformers have been extended. The synthesis procedure is based upon approximate solutions to the equations describing nonuniform transmission lines that have appeared in the literature. Using these approximate solutions, the matching problem is reduced to the problem of finding a real function, the reflection-distribution function, whose complex Fourier transform approximates a complex function, determined by the generator and load impedances. The reflection-distribution function must be identically zero outside a specified interval. The reflection-distribution function is found by expansion into a trigonometric series and subsequently determining the coefficients in this expansion. A method is developed by which the complex function to be approximated is first separated in real and imaginary parts. The coefficients in the trigonometric expansion are then determined such that these real and imaginary parts are approximated separately in a discrete Chebyshev sense. By discrete Chebyshev sense is meant that the maximum magnitude of the approximation error is minimum at a discrete number of sampling points. The approximation process makes use of the theory of discrete Chebyshev approximation. The theory of discrete Chebyshev approximation, subject to constraints, has been treated. The constraints arise from the necessity to control the characteristic-impedance level at the terminals of the uniform transmission line. The result of this investigation is a general synthesis procedure which extends the methods presently available. Not only matching between real impedances, but also matching between complex impedances can be achieved. Several examples have been given of the synthesis of impedance transformers and matching sections. xi

CHAPTER I INTRODUCTION The subject of nonuniform transmission lines has attracted considerable interest with the development of microwave techniques during recent years. Synthesis of conventional networks becomes increasingly difficult at high frequencies because of the complications caused by parasitic elements. Nonuniform lines offer a very attractive solution to this problem, since the upper frequency is limited only by the occurrence of higher modes when the transverse dimensions of the line are comparable with the wavelength. An exact solution of the differential equations describing the nonuniform line is'possible in only a few special cases. This has led to the development of a number of approximate solutions. The most widely used of these is the method developed by Bolinder (Refs. 2, 3), because it provides the best approximation with regard to accuracy and simplicity. The requirement, however, that the nonuniform line be matched at the receiving end, has restricted the use of Bolinder's method to the synthesis of impedance transformers. An impedance transformer is a nonuniform line that provides a match between two real impedances of different value. Recently, Orlov (Ref. 9) and Sharpe (Ref. 11) have developed an approximate solution which is valid when an arbitrary mismatch exists at the receiving end of the line. Their solution makes it possible to consider the synthesis of matching sections for which both the load impedance and the internal impedance of the generator are complex quantities. 1

2 The subject of this dissertation is the development of a synthesis procedure based upon the approximate solution by Orlov and Sharpe. The procedure involves the construction of a bounded real function, which is identically zero outside a specified interval, whose complex Fourier transform approximates a given complex function. The theory of discrete Chebyshev approximation will prove to be a very powerful tool in this approximation process. The present method is particularly suited for the synthesis of nonuniform lines whose behavior must be controlled over a given range of frequencies. The method is completely general and can therefore also be applied to the synthesis of impedance transformers and driving point impedances. The -treatment in this dissertation is subdivided into several chapters. The next chapter, Chapter II, contains a brief statement of the problem. In Chapter III a review is given of the pertinent literature. The literature is subdivided into two principal categories. In the first the approximate solutions to the differential equations of nonuniform lines are developed, in the second, some of these approximate solutions are applied to the synthesis of impedance transformers. In Chapter IV, the general matching problem is considered and reduced to an approximation problem, which can be solved using the theory of discrete Chebyshev approximation, as developed in Chapter V. Chapter VI is devoted to examples of the synthesis of impedance transformers and matching sections. Conclusions are given in Chapter VII.

CHAPTER II STATEMENT OF THE PROBLEM The principal problem considered in this study is the synthesis of matching sections. By a matching section is meant a nonuniform line, of finite length, which can be inserted between a generator and a load to obtain maximum energy transfer over a given range of frequencies. Both the internal impedance of the generator, Z1, and the load impedance, Z2, are complex quantities, which are functions of frequency. The circuit configuration is shown in Fig. 2.1. Zo(Y) L, I LOAD GEN (- Z2 y=O y y= NONUNIFORM LINE -- Fig. 2.1 Circuit configuration for the matching problem. The synthesis problem can now be stated as follows: given the internal impedance (Zl) of the generator, the load impedance (Z2), and the frequency range over which they are to be matched, synthesize the matching section. The matching section that will be synthesized is a section of nonuniform transmission line subject to the following restrictions: 3

a. The length, 2, of the nonuniform line is finite. b. The nonuniform line is lossless. c. The taper is continuous. The last restriction is equivalent to the requirement that the characteristic-impedance function, Zo(y) be continuous. It will be assumed that Zo(y) does not go to zero or infinity at any place in the line. The synthesis is completed when the characteristic-impedance function has been determined. The characteristic-impedance function, in turn, is uniquely determined by the reflection-distribution function, P(y), which is defined by the relationship p(y) = 1 (Zo) 1 d In Zy(2.1) 2 ZO (y) dy 2 dy Because of the requirement (b) that the nonuniform line be lossless, the function p(y) is a real function. Because of the requirement (a) that the nonuniform line is of finite length, the function p(y), for mathematical convenience, is defined to be identically zero outside the interval (0,1). The requirement that the taper be continuous implies that the function p(y) is bounded. The synthesis problem can be reduced to finding a function p(y) such that its complex Fourier transform, G(s), approximates a complex function rl(. The function rl(s), as will be shown, can be determined from the impedances Zl(s) and Z2(s), which are given. The variable s is defined by s = ~/%, where 2 is the length of the matching section and X the wavelength. The variable s, therefore, is dimensionless and proportional to frequency. It will be called the frequency

variable. In general G(s) can only approximate rl(s), because not every complex function rl(s) is the Fourier transform of a real function which is zero outside the interval (0,1). The function p(y) will be expanded in a trigonometric series: N p(Y) = Z [an cos nty + bn sin nay] (0 < y < 1) n - n(2.2) ply) = O (y < O; y > 1) and the coefficients a and b will be determined such that ReIG(s)ei2 <I n n approximates Re {rl(s)ej2's}, and Im{G(s)e02 s}approximates Im rl(s)ej2s} in a discrete Chebyshev sense. A function f(s) is said to approximate a function F(s) in a discrete Chebyshev sense when the maximum of the magnitude of the error max Ih(s)l = max JF(s) - f(s)l (2.3) is minimum at a discrete set of sampling points. Once the values of the coefficients a and b are known, the n n reflection-distribution function p(y) and, therefore, also the characteristic-impedance function ZO(y) are completely determined. The determination of the coefficients a and b, therefore, essentially comn n pletes the synthesis of the matching section. Other problems that will be considered are special cases of the general matching problem that was outlined above. These are the synthesis of impedance transformers (Z1 and Z2 both real) and the synthesis of a driving point impedance which approximates a given impedance over a range of frequencies.

CHAPTER III REVIEW OF THE LITERATURE 3.1 Introduction The work on continuously tapered lines can be subdivided into two principal categories. To the first belong attempts to find solutions, exact and approximate, to the differential equations describing the behavior of a nonuniform line. The second category consists Of synthesis procedures based on these approximate solutions. The approximate solution that has played a major role in the synthesis of nonuniform lines is the Fourier transform solution. In short, it states that the input reflection coefficient of a nonuniform line equals the Fourier transform of the reflection-distribution.function. This solution was proposed by Bolinder (Refs. 2, 3) and was used by many others to synthesize nonuniform lines to act as impedance transformers between real impedances. The reason that only this kind of impedance transformer has been synthesized lies in the fact that the Fourier transform approximation is valid only when the line is properly terminated at the receiving end. Recently Orlov (IRef. 9) has developed an approximate solution that is valid when an arbitrary mismatch is present at the receiving end of the line. This solution was independently derived by Sharpe (Ref. 11) using a different approach. In the following paragraphs a review of the pertinent literature will be given. The symbols used by the authors in their original work have been modified, where necessary, to arrive at a uniform notation throughout this review. 6

7 3.2 The Differential Equations for Nonuniform Lines The basic assumption that is always made in the treatment of nonuniform lines is that of the existence of a unique current I(x) and voltage U(x), which at any point in the line satisfy the equations: d(x) = - y(x)U(x) (3.1) dx dU(x) = - z(x)I(x) (3.2) dx where y(x) is the shunt admittance and z(x) is the series impedance per unit length. These equations are valid only when certain conditions are imposed on the electromagnetic field and the line: (a) the mode of wave propagation must be essentially transversal (TEM), which implies that all wavelengths must be large in proportion to the transverse dimensions of the line; (b) there should be no rapid discontinuities in characteristic impedance. By eliminating U(x) or I(x) from Eqs. 3.1 and 3.2, two secondorder differential equations can be obtained. d2 (x)d y(x) dI(x) _ y(x)z(x)I(x) = O (3.3) dx2 dx dx d2(x) d ln z(x) dUx) y(x)z(x)(x) = (34) dx2 dx dx When y(x) and z(x) are constant along the line, Eqs. 3.3 and 3.4 reduce to the well-known equations foar uniform transmission lines. In general, however, (3.3) and (3.4) cannot be solved except in some special cases

such as the exponential line. It is possible to reduce the order of the differential equations (3.3) and (3.4) by making the proper transformations. The following quantities are defined' The characteristic-impedance function: Zo(x) - y (3x5) The propagation-constant function: Y(x) - o/z(x) y(x) (3.6) The reflection-coefficient function: U(x) (X) kx) 7' (3 7) -xx) + Zo (X) Consider the identity: U(x) d I~ 1. U(x) (3(x) dx dx I(x)2 dx Substituting (3.1) and (3.2) into (3.8), one obtains d U(x) (x) - z(x) + y(x) [(xj (3 9) From (3.7) it follows that = Z0(x) e (3.10)

9 Substitution of (3.10) into (3.9) yields 1 + (x) dZ0 ZO dF(x) dp(xj X [-(()2 +jl -(x)] dd + r+(x) ] d (3.11) - z(x) + y(x) Zo()2 [1 +(x)] 2 Using (3.5) and (3.6) and rearranging terms, (3.11) reduces to 1 2 2 d I (x) -r 2 7(x)5(x) +21 [- r(x)2] d lnZ(x) = O (3.12) dx 2 dx The reflection-distribution function is now defined as follows: d in ZO(X) p(x) _ (3.13) 2 dx The differential equation (3.12) can then be written as follows: d-x - 2 7(x)5(x) + [1 - F(x)2 ] p(x) = O (314) dx This is the differential equation in reflection coefficient for a nonuniform line. It is a first-order nonlinear equation known as a Riccati equation. It is exact, and if it could be solved it would give exact solutions for the problem of nonuniform lines. In the following paragraphs the approximate solutions to Eq. 3.14 will be presented. 3.2.1 The Approximate Solution for Terminated Lines. Bolinder (Ref. 3) proposed to study those nonuniform lines for which ir(x) 12 << 1, everywhere on the line. This implies that there be no mismatch at the receiving end of the line. With this assumption, the term 4 r(x) can be neglected in Eq. 3.14. One then obtains an approximate differential equation:

O10 dX d. 7(x) - 2 Y(x) F(x) + p(x) 0 (3.15) This is a linear first-order differential equation. It can be solved x using an integrating factor, exp [-2 f y(S) d]s, in which the lower a limit of integration is arbitrary. The solution that satisfies the boundary conditions is: -2 f 7(e) da r(x) = f p(>) e Xd (3.16) x The reflection coefficient at the input of the line equals x -2 f y(S) di r 7 r(o) f p(x) e ax (3.17) For lossless lines r(x) = jp(x). When the dielectric in the line is homogeneous, r(x) is constant along the line:,(x) = jo = j (3.18) and (3.17) becomes = fp(x) e-ji x dx (3.19) 0 The restriction that the dielectric be homogeneous can easily be removed. For the case in which the dielectric is not homogenous, an expression equal to (3.19) can be obtained in which the variable x has been replaced by a new variable u, and the constant 3 by a new constant B'. The variable u and the constant At are defined by the relation

11 x t'u = f P(x)dx. Introducing this variable u amounts to measuring 0 the distance along the line in wavelengths. To simplify the form of expression (3.19), a normalized coordinate, y, is introduced: X (3.20) where ~ is the length of the nonuniform line. A new frequency variable, s, is introduced also: s = " (3.21) It is readily seen that s is a dimensionless variable, proportional to frequency. With these new variables (3.19) can now be written in its final form. r(s) f= p(y) eJ4sY dy (3.22) This, then, is the well-known Fourier transform approximation for the input reflection coefficient. The limits of integration on the Fourier transform are -ao and +w. Because, however, the function p(y) is identically zero outside the region of the nonuniform line, i.e., p(y) = O outside the interval (0,1), Eq. 3.22 does represent the Fourier transform of the reflection-distribution function p(y). It should be pointed out that (3.22) can also be derived by a more direct approach. One can argue that the reflection generated by a portion, dx, of the line, located at the point x, can be written as Zo(X dx) - Z0(x) p(x) dx Z0(x+dx)+ Z0(x)

12 A portion of length dx of the line, located at the point x, therefore contributes an amount dFto the total reflection at the input of the line, which can be written as: dr = p(x) dx e-j2px The total input reflection is then found by integrating the contributions from all points along the line. The result is Eq. (3.19). The Fourier transform has become the most widely used and most convenient tool in the synthesis of nonuniform lines. Due to its restrictions, small reflections and matched load, the synthesis effort has been confined to nonuniform lines acting as impedance transformers between real impedances. Some of the representative procedures will be reviewed in section 3.3. 3.2.2 The Approximate Solution in the Presence of Load Reflections. Recently Orlov (Ref. 9) and Sharpe (Ref. 11) have developed an approximate solution that is valid for the case in which an arbitrary mismatch exists at the receiving end of the line. Orlov considers the behavior of a line varying in small discrete steps. By letting the number of steps go to infinity, Orlov obtains a solution for a continuously varying line which has an arbitrary mismatch at the receiving end. Let the reflection at the load be equal to r2: Z2 - Z02 r2 z.+ Z (3.23) where Z2 is the value of the load impedance and 02 is the value of the characteristic impedance of the line at its receiving end.

13 The load reflection can be referred back to the input of the line. This defines the quantity ro, which can be written as: r1 = r2 e'j25 = r2 ej45s (3.24) Orlov obtains a complicated expression for the input reflection coefficient, which in first approximation can be written as follows: 1 S -+ Y) - j4e sydy po + f p(y) e3 dy r(s) = 0 (3.25) 1 + r p(y)() en dy 0 When the line is matched at the receiving end (ro = 0), this expression reduces to the familiar Fourier transform (3.22). Sharpe treats the problem of the nonuniform line, terminated in a mismatch, as a one-dimensional scattering problem, using perturbation techniques. An expression for the input reflection coefficient is then obtained in the form of a Fredholm series expansion. The firstorder approximation to this series expansion agrees with Orlov's result (3.25). According to Orlov it is required that IP(Y)I << 1. Acmax cording to Sharpe the condition IP(y)I << 2ns should be satisfied. max From (3.25) both Orlov and Sharpe derive a synthesis formula, which gives the relationship between the reflection-distribution function p(y) and the reflection coefficients at the input and output of the nonuniform line: 1J P -j4,sy r(L -ro ) - o( - Irl)'p(y)e dy = I.... ) - "p ) (3.26) 0 F2171

14 This expression will be the basis for the discussion in Chapter IV, where the synthesis of matching sections is developed. 3.3 Applications of the Fourier Transform As was mentioned above, the synthesis of nonuniform lines has been restricted so far to impedance transformers (between real impedances) because of the requiremelEnt that the line be properly terminated at the receiving end. In this part of the literature review several of these synthesis methods and their results will be discussed. 3.3.1 The Synthesis of Impedance Transformers. A method to synthesize impedance transformers was presented by Willis and Sinha (Refs. 13, 14). The same method was described by Baur (Ref. 1), who obtained essentially the same results. In this method the reflection-distribution function p(y) is expanded in a trigonometric series containing either odd sine terms or even cosine terms. The input reflection coefficient fI(s)| is then determined using Eq. 3.22. Let p(y) be expanded in a sum of odd sine terms: N p(y) = Z b sin nty (3.27) n=l where n is an odd integer and the b's are constants. Evaluating the n integral (3.22) one finds the input reflection coefficient of the line: If(s)l = z. b 2n cos 2es (n odd) (3.28) n=l n d[(4s) n2] The reason given by Willis and Sinha for choosing only odd values of n is the fact that only odd. sine terms contribute to impedance

15 transformation. The amount of impedance transformation is found by integrating p(y). It follows from the definition of p(y), Eq. 3.13, that 1 1 Zo(L) f p(y) dy = In v() (3.29) 0 Therefore the amount of impedance transformation caused by even sine terms is zero because 1 f sin nay dy = 0 when n is even. 0 The amount of impedance transformation caused by the odd sine terms is equal to: 1 1 2 f sin niry dy- = n- cos ncy = - for n odd. (3.30) 0 There is, however, a more important reason for using only odd values of n, which is not mentioned by Willis and Sinha. The contributions to the input reflection coefficient by the odd sine terms are all in phase and 90 degrees out ofphase with the contributions from the even sine terms. Using only odd values of n therefore has the advantage that the input reflection coefficient Ir(s)l can be written in the form of a simple addition as expressed in Eq. 3.28. Willis and Sinha also consider the case in which p(y) is written as the sum of even cosine terms. Written in this form: N p(y) = Z an cos ngy (n even) (3.31) n=0O

16 The input reflection coefficient IF(s) resulting from this reflectiondistribution function becomes N 8s sin 2is n=O n c[(4s2- n2] if(s~,| n=0 an [(4 )2 -S ] (n even) (3.32) Using either (3.28) or (3.32) Willis and Sinha then proceed to determine the coefficients bn and an to obtain a high-pass characteristic for Fr(s) 1, with a minimum amount of reflection in the pass band. The coefficients in their method are determined by trial and error. Figures 3.1 and 3.2 show two reflection patterns obtained by this method. Figure 3.1 shows the reflection pattern for an impedance transformer that is 0.75 hX long at the lowest frequency of the pass band. The corresponding reflection-distribution function is equal to p(y) = k(l - 0.636 cos 2 y) (3.33) Figure 3.2 shows the reflection pattern for a line which is 1 X long at the lowest frequency of the pass band. The p(y) producing this pattern is proportional to: p(y) = k(l - 0.899 cos 2Ty + 0.0112 cos 4iy) (3.34) The maximum height of the side lobes is 0.031 for the 0.75 X line and 0.0056 for the 1 X line, for the case in which k = 1. The proportionality factor k is selected to give the correct amount of impedance transformation. For lines characterized by (3.31), where only the constant term gives impedance transformation, the factor k is determined by:

17 1.0.8 Irl T.6.4 Fig. 3.1 Reflection pattern for p(y) = 1 - 0.636 cos 2ity..3 -.2.05 1.0.005 0.5 1.0 1.5 2.0 2.5 - S Fig. 3.2 Reflection pattern for p(y) = 1 - 0.889 cos 2ty + 0.0112 cos 4~y.

18 k = Zo(l) k = In v, ) (3.35) Everywhere in the present discussion k will be taken equal to k = 1. When the line is expanded in odd sine terms, (3.27), k' must be determined from the following equation, using the result (3.30): Zo1) N b In k' EZ _n (n odd) (3.36) z0(o) n=l nt Measurements were taken by Willis and Sinha on impedance transformers synthesized by this method. Excellent agreement between theory and experiment is reported in Ref. 14. Very similar work was reported by Feldshtein (Ref. 6), although his method is less general. He considers a nonuniform line for which the reflection-distribution function p(y) can be written in the form p(y) = k(l + a2cos 2iry) (3.37) Somewhat arbitrarily he defines this function to represent an optimum smooth transition. Feldshtein then proceeds in a manner similar to that of Willis and Sinha and finds that minimum reflection in the pass band is obtained for a2 = - 0.632 in the case of a 0.75 X line and a2 = - 0.840 in the case of a 1 A line. The height of the side lobes thus obtained is equal to 0.032 for the 0.75 X line and 0.0082 for the 1 X line. Feldshtein's result for the 0.75 X line is essentially equal to that obtained by Willis and Sinha. His 1 X line gives slightly more reflection in the pass band, because only two terms are used in the

19 expansion for p (y). Another example of the use of the Fourier transform has been demonstrated by Klopfenstein (Ref. 7). He reminds his readers of the analogy between the uses of the Fourier transform in transmission lines and in antenna pattern design, an analogy that was already pointed out by Bolinder (Ref. 3). Klopfenstein then adapts to transmission lines the work of Taylor, who studied the synthesis of Dolph-Chebyshev patterns using continuously illuminated apertures. This procedure leads to an input reflection pattern consisting of a main lobe and infinitely many side lobes, all of equal height. The taper that produces this reflection pattern is characterized by two step discontinuities in Z(y), one at the beginning of the line and one at the end. The height of the side lobes equals cosh (2 s0) where s is 0O the value of s = - at the lowest frequency of the pass band. Thus it is found that the maximum reflection in the pass band equals 0.018 for the 0.75 X line and 0.0037 for the 1 X line, results that are significantly better than those obtained by Willis and Sinha. 3.3.2 Nonuniform Lines as Filters. Feldshtein (Ref. 5) studied and experimentally measured the behavior of lines whose tapers vary rather violently. He shows that these lines exhibit the properties of a band-rejection filter. In the theoretical derivations, the Fourier transform approximation is used. Consider a line for which the reflection-distribution function is: p(Y) = 3.3 cos 2ty (3.38) The ratio between the maximum and minimum value of the characteristic

20 impedance [ZO(y)] of such a line is equal to 8. When the Fourier transform of p(y) is taken an input reflection IF(s)l is obtained, whose absolute value exceeds unity. Since it is impossible for a reflection coefficient to exceed unity, Feldshtein defines Fr(s)I to be equal to one for those values of s, for which the absolute value of the Fourier transform of ply) exceeds this value. The reflection coefficients obtained by this process are shown in Fig. 3.3 and in Fig. 3.4 for the case in which p(y) = 6.6 cos 4xy. The broken lines in these figures show the results that were obtained experimentally by Feldshtein. It is interesting to note that the experimental values are reasonably close to the calculated values. This is unexpected because the Fourier transform approximation is valid only whenr Jf2 << 1, a condition that is by no means fulfilled in this example.

21 I.5O 0 0.5 1.0 1.5 2.0 Fig. 3.3 Theoretical and experimental filter behavior for p(y) = 3.3 cos 2ry t'57 Alx \ X I I I,,5 i.. 1.5 2.0 0 0.5 1.0 1.5 2.0 -"*S Fig. 3.4 Theoretical and experimental filter behavior for p(y) = 6.6 cos h4y.

CHAPTER IV THE SYNTHESIS OF MATCHING SECTIONS 4.1 Introduction In this chapter the synthesis of matching sections will be developed. By matching section is meant a nonuniform line which can be inserted between a generator and a load, to obtain maximum energy transfer over a certain range of frequencies. Both the output impedance of the generator and the load impedance are complex quantities, which are functions of frequency. The synthesis method developed in this chapter is based on the approximate solution developed by Orlov (Ref. 9) and Sharpe (Ref. 11). The matching sections that are synthesized with this method are sections of nonuniform line, which have the following properties: a. The length of the nonuniform line, Q, is finite b. The nonuniform line is lossless and has a homogeneous dielectric c. The taper is continuous. It will first be shown how the formula developed by Orlov can be used to reduce the matching problem to the problem of finding a complex function rl(s) whose inverse Fourier transform is the function p(y), the reflection-distribution function. The requirement that the nonuniform line be lossless and of finite length puts restrictions on the function p(y). p(y) must be a real function and identically zero outside the interval (0,1). In general the inverse Fourier transform of rl(s) will not meet these conditions. A complex valued function G(s) will therefore be sought which approximates rl(s). The inverse Fourier transform of G(s) will be a real function, zero outside the interval 22

23 (0,1). The error in the approximation will cause a slight mismatch, giving rise to undesired reflections. In the synthesis procedure these undesired reflections are minimized. 4.2 The General Synthesis Problem When a complex load impedance is connected to a lossless transmission line, there will always be a reflection from this load, because the characteristic impedance of the line is real. Specifically, if the line is a lossless nonuniform line, an approximate solution is required that is valid in the case that such load reflections exist. As was discussed in Chapter III, such an approximate solution has been recently developed by Orlov (Ref. 9) and Sharpe (Ref. 11) (see section 3.2.2): F(~ -I1ot2) - Fo(l - 2Fi ) F[p(y)] -= p(y) e-j4s5Ydyy = o (4.1) 0 10 where p(y) is the reflection-distribution function: 1 d In Zo(y) p(y) = (4.2) 2 dy F, which is a function of the frequency variable s, is the input reflection coefficient of the nonuniform line and r is the load refleco tion (r2) referred to the input of the line: Fo F2 e-j4ts (4 3) Furthermore the frequency variable s equals: s = -, i.e., the ratio between the length of the nonuniform line and the wavelength. Therefore s is proportional to frequency.

24 The function Fr(s) will next be defined. It is equal to the right-hand side of Eq. 4.1: F1(s) -Ir) (4ro-t4)2 The circuit configuration is illustrated in Fig. 4.1. Maximum power transfer is to be obtained from the generator, whose internal impedance is equal to Z1, to the load (Z2). This matching is to be achieved using a nonuniform line of finite length. The line extends from y = 0 to y = i, as shown in Fig. 4.1. Zol Zo(y) Z02 LOAD GEN. 2 y =. I k-NONUNIFORM LINE Fig. 4.1 Circuit configuration for the matching problem. The condition necessary to obtain maximum power flow at the input of the line is well-known. The input impedance of the nonuniform line, terminated in Z2, must equal Z1, the complex conjugate of Z1. This, of course, guarantees maximum power flow only across the point y = O. However, together with the fact that the line is lossless, it also implies maximum power transfer from generator to load.

25 Let the characteristic impedance at the input of the nonuniform line be equal to Zo0() = ZO1. When the input impedance of the line is equal to Z1, the input reflection coefficient (F) will be: Z1 - Z01 1= 0......(4.5) Z1 + ZO 1 01 Let the characteristic impedance at the receiving end of the line be equal to Zo(1) Z02. The load reflection (r2) is then equal to Z2 - Z02 2 = 2+ 02 (4.6) s - s2+ Z02 r is the load reflection referred back to the input of the line, so 0 that z' -z 1' Z + e -jhrs (4.7) 2 + Z02 where 17, r2, Io, Zl, and Z2 are in general functions of frequency. The discussion can now be continued as follows. Let the configuration of Fig. 4.1 be given where the characteristic impedance of the nonuniform line at the input and at the receiving end is equal to Zol and Z02' respectively. The function ro(s) is then determined, and the function r(s), necessary for maximum power transfer, can be determined from Eq. 4.5. The functions r(s) and F (s) completely determine the function rl(s), as defined by Eq. 4.4. It seems that by taking the inverse Fourier transform of rl(s), one could then find the reflection-distribution function p(y):

26 p(y) = F [ li(s)] 2 f cl(s) ej"'f ds (4.8) Unfortunately the situation is not that simple because of the restrictions imposed on p(y). These restrictions arise from the requirement that p(y) be the reflection-distribution function of a lossless line of finite length. p(y) must be a real function which is identically zero outside the interval (0,1). In general the inverse Fourier transform of rl(s) will not yield such a function p(y). In the following sections a method will be developed by which a function G(s) can be found which approximates the function rl(s) in a Chebyshev sense over a given interval of frequency, at a number of discrete sampling points. The most important property of G(s) is, that its inverse Fourier transform is a function p(y) which is real and identically zero outside the interval (0,1). The error E(s) made in this approximation process will be: E(s) = rl(s) - G(s) (4.9) E(s) equals the amount of undesired reflection between generator and load when they are connected by the matching section. 4.2.1 The Case of a Real Generator Impedance. Before the discussion is continued it will be of interest to consider as a special matching problem the case in which the generator has a real internal impedance (Z1 = R1), independent of frequency (see Fig. 4.1). By choosing the characteristic impedance of the line at its input equal to R1 (ZO = Z0(0) = R1), maximum power transfer is obtained when r = 0. The synthesis formula (4.1) then reduces to:

27 F[p(y)] = - F (4.10) It is interesting to note the physical significance of this result, which can also be arrived at using the following consideration. Consider a nonuniform line matching a generator to a load impedance Z2 as shown in Fig. 4.1. The characteristic impedance (Z01) Of the line at the input equals R1, the internal impedance of the generator. For maximum power transfer, the input reflection coefficient of the line must be zero. The total input reflection of the line is principally generated by two sources. One is the reflection from the load (r2). Referred back to the input of the line, the load contributes an amount, equal to o~, to the input reflection coefficient (Eq. 4.3). The other source is the integrated reflection from the nonuniformities of the line. This contribution can be expressed by F[p], the Fourier transform of the reflection-distribution function. The input reflection coefficient of the line will be zero when these two-reflections exactly cancel each other. This occurs when they are equal in magnitude and 180 degrees out of phase. In other words, matching is achieved when F[p(y)) = - (40) which is exactly the same result that was obtained above. 4.2.2 Synthesis of Driving Point Impedances. The synthesis of driving point impedances can be reduced to the same problem that arises in the synthesis of matching sections, i.e., to find a real function p(y) which vanishes outside the interval (0,1) such that its

28 Fourier transform approximates a given complex valued function F(s). To synthesize an impedance that behaves like a given impedance Z(s) over a certain band of frequencies, it is sufficient to synthesize a function f(s) such that Z(s) - Z r(s) = Z(s)+ Z0o (4.1) where r(s) if the input reflection coefficient of a nonuniform line of finite length, terminated at the receiving end. The problem then becomes to find a function G(s) that approximates F(s). The inverse Fourier transform of G(s) is a real function p(y) that is identically zero outside the interval (0,1). 4.3 The Determination of the Reflection-Distribution Function In the previous sections the synthesis problem was reduced to the problem of finding a function G(s), which is the Fourier transform of the reflection distribution function p(y) for a lossless nonuniform line of finite length. From the assumption that the line is lossless, it follows that ply) is a real function, because the characteristic-impedance function Z0(y) is real. From the requirement that the line be of finite length, it follows that the function p(y) must be identically zero outside the interval (0,1). In this section the nature of the function G(s) will be determined. It will be shown that a convenient expression for G(s) is obtained when the function p(y) is expanded in a trigonometric series: p(y) = Z [a cos nry + bn sin nty] (4.12) n=0 n

29 One is in general interested in letting the matching section provide a match over only a finite range of frequencies, and therefore a finite number of terms will be used in the expansion (4.12). Of course p(y) is identically zero outside the interval (0,1). The function G(s) is the Fourier transform of p(y) and is therefore equal to: N G(s) = F[p] = a { an F[cos nity] + bn F[sin niy]} (4.13) n=O The coefficients an and bn must then be determined in such a way that the function G(s) is a good approximation to the function rl(s). In the following paragraphs the Fourier transforms of cos niy and sin nity will be evaluated. 4.3.1 Evaluation of F[cos niy]. The Fourier transform of cos nty- is equal to F[cos nTy] = cos niy e 4ASYdy (4.14) 0 This integral can be calculated by first integrating by parts twice. ~1 4 i 1 2 1 sni4s-j4s e jssy eJ4tYdy isin niy ejisy e cs nity ) + cos..y n 0 n 0 n 0

30 This expression can be reduced to [ ()2 ] J cos nsty e 4sd [l cosne n 0 n v Because cos nt = (-1)n this can be written as: 1 i2 _____________ _ 4sn f cos ny ej-nsYdy j 4s [1 -(-1) e-j4s] 0 n - (4s)2 ] A factor e j2is can be brought outside the brackets in the right-hand side of this equation, resulting in: 1I -j4gsy j4s e-j27cs [e2it ()n e32its f cos nty e y = [ej (-)n - 0 g[ 2 (4s) 2 (4 (4.15) For even values of n, the term inside the brackets reduces to [ej2s - ejs ] = 2j sin 2is For odd values of n, the same term reduces to: [ j2s + e-j2s ] = 2 cos 2xs This completes the evaluation of the integral (4.14) which can now be written in final form: -8s sin 2is -j2is 1J 2cos~~ 2n~ ~e for n even cos niy ej4SYdy it[n (4s) (4.16) 0 j8s cos 2is -j2s or n odd [n2 - (4s)2]

31 To achieve convenient expressions in the following paragraphs, *the following notation will be introduced. Even values of n are designated by ne, odd values by no. The functions C (s) and Cno (s) are now defined: -8s sin 2~ts C (S) -8s sin 2vs (n even) (4.17) Cne 4tn - (4s) ] C (s) 8s cos 2vs n - 2s] (n odd) (4.18) n"o Tn2 _~ (4s) Or written in slightly different form: Cne (s) I 4s n Cne(s)= sin.. + 2 [s+ 11 (n even) (4.19) Cno(s) = -cos 2s [ n + 1 (n odd) (4.20) no[4s + n +4sUsing the notation adopted above, the Fourier transform of a trigonometric cosine series can now be written as follows: N N N 1 F Z a cos nry a neC (s) + j no a Cn(s es Ln=O j n=O n=l (4.21) As an example, the function C4(s) is plotted in Fig. 4.2. The functions C (s) equal zero whenever 4s assumes an even ne integer value, different from ne. When 4s approaches ne, Cne(s) approaches a limit that can be evaluated using de l'Hpital's rule. lim C (s) = lim -8s sin 2is 4s-*n ne 4sn [n2- (4s)2] -8 sin 2xs - 16is cos 2ts =slim...... -3 s 4s -- n

32.5.4.2 Fig. 4.2 Plot of the function C4(s). The denominator is non-zero when 4s approaches n, provided that n O0. Because n is even: sin 2its = 0 when 4s = n and cos 2s = cos (-1)/2 when 4s = n. Therefore lim Cne(s) 2 (-1)n/2 for ne O (4.22) 4s - n For the case in which n = O, de l'Hrpital's rule has to be applied a second time. lirna CO (s) = l irna-8 sin 2rrs - 16gs cos 2is 4s -;rO 0s-*O -32..s -16 cos 2ts - 16rcos 2cs + 3272s sin 2..s lim -..... 4s - o0 -32=

33 Therefore lim CO(s) = 1 (4.23) 4s-40 Similarly, the function C (s) = 0 whenever 4s assumes an odd no integer value different from no. When 4s approaches no, C n(s) approaches a limit whose value can be found using de lt'rpital's rule. 8s cos 2gs lim Cno(s) lim 4s s-n o4s -n [n _ (4s)2] lim 8 cos 2its - 16ts sin 2vs 4s -* n -32irs The denominator does not vanish when 4s approaches n. Because n is odd: cos 2gs = 0 when 4s = n nT (n-1)/2 and sin 2gs = sin - = (-1) when 4s = n. Therefore lim Cno(s) = 2 (4.24) 4s -+n 4.3.2 Evaluation of F[sin nty]. The Fourier transform of sin ngy is equal to 1 F[sin ngy] = f sin nity eitsydy (4.25) 0 Like F[cos niry] it can be evaluated by integrating by parts twice.

34 f sin nty e-4szay - 0 " cos0 n~y,-j4gsy - _ j4s f cos nity e d = nit r~0 n 0 -j4rcsy 1 2 1 -1y e sin nqy e sin n y -j4sy + 16s2 ity.e C o 0 n n o This expression can be reduced to: 1 - (4s) ] f sin niy e-j4SYdry 1 [1 - cos n e-j4is] n 0 Because cos ni = (-1)n, this can be written as: f sin nity e 45SYdy [1 2 _( )n ( -j4es] ~O~ ~[n - (4s) 3 Proceeding as before, a term e j2is is eliminated from the brackets on the right-hand side, giving: ~1 -j4isya _ n e'jis, j2is (sn -j2gs f sin nity e 42SYdy [eiS (1)n e2] O [n - (4s) ] (4.26) For even values of n, the expression inside the brackets becomes [ej2is - e-j2is] = 2j sin 2ris For odd values of n, the same expression reduces to [eJ2is + e-j2S ] = 2 cos 2is

35 This completes the evaluation of the integral (4.25). Written in final form: j2n sin 2-rts -j2s for n even t[n2 _ (4s) ] Using again the notation ne and no for even and odd values of n, respectively, the functions S ne(s) and Sn (s) can be defined: 2n sin 2s(4.2),net(S)= 2_ ( 2s) 2 (n even) (4.28) S (s) 2n cos 2irs (n odd) (4.29) [n2 _ (4s) ] These functions can also be written in slightly different form: Sne(s) = -sin 2"'s +i n 4s - n (n even) (4.30) So (S) cos2s 27 4s 1 4s. (n odd) (4.31) no 4s + no4[ n With this notation, the Fourier transform of a trigonometric sine series can be written in the following form: n=1 [ n=l n_2 (4-32 F nl in 1 nl noSno(S) + i L bneSe (s)j eAi2s(432) The function S5(s) is plotted in Fig. 4.3 as an example. As is apparent, the functions S,(s) behave essentially like the functions Cn(s). The functions Sne(s) equal zero whenever 4s assumes an even

36.4 S5(s).5 1.0 1.5 2.0 2.5 S -,1 -.2 Fig. 4..3 Plot of the function S5(s). integer value, and the functions Sno(s) are zero when 4s assumes an odd integer value, except in the case that 4s = ne and 4s = no, respectively. The limit of Sn(s) when 4s approaches n can again be evaluated using the rule of de l'H Cpital. lim Se(s) = lim 2n sin 2)s 4s-,n 4s-,n [n2 - (4s)] lim 4n cos 2As 1 nA 4s -n 0-32its 2cos 2 (4.33) (- (l l+(n/2) (n even) S imi larZly

37 2n cos 2its lim S (s) 3 lim 2n cos 2s 4s-n no 4s-*n [n - _(4s) 2 limr -4itn sin 2ts = 1 sin n- (4.34) 4s — n -32is 2 2 1 (-I)(n-1)/2 (n odd) 4.3.3 Impedance Transformation. Because the functions F(s) and rF(s), which arise in the synthesis problem, depend on the value of the characteristic impedance of the nonuniform line at its terminals, it is of importance to know how Z01 and Z02 are affected by the reflection-distribution function p(y). From the definition of p(y): d in Z (y) p( =) 2 dy (4.2) it follows inmmediately by integration that 1 1 Zo(l) 1 Z2 f p(y) dy 2 in - 2 n (435) 0 Z0I() 01 When p(y) is expanded in a trigonometric series, N p(y) = Z [ancos nty + b sin niy] (4.12) n=O the amount of impedance transformation along the line can be expressed in terms of the coefficients a and b n n First the amount of impedance transformation caused by the cosine terms, is evaluated:

38 J1 d, sin ny;1 sin 0 for n r 0 cos nicy dy= - in nity I 0 nt 0i = I for n =O (4.36) Thus it is seen that cosine terms do not contribute to impedance transformation, except when n -= 0, which corresponds to the constant term in the expansion. Secondly the amount of impedance transformation contributed by the sine terms is evaluated: 1 L 1 - cos ni f sin niy dy = - cos nty I = (4.37) 0 nit0 n -(-K7O for n even nit 2 f or n odd nxn The amount of impedance transformation resulting from a reflection-distribution function p(y) given by Eq. 4.12, is equal to: z N 1in 02 2 n z0 = a0+ b no no2 (4.38) 2 Z01 n=l 4.3.4 Surmmary of Results. The results obtained in Sections 4.3.1 and 4.3.2 can now be combined into a single equation. When the reflection-distribution function p(y) is given by the trigonometric expansion (4.12), its Fourier transform F[p] can be written as follows: F[pl = G(s) f= p(y) eJ4Ydy = 0 N n=0[ane Cne(s) + b S (s)]+ j Z [a Cn (s) + b (s) e-j2xs ne ne ~n no n=l no on ne (4.39)

39 There now remains the determination of the coefficients a and b. The n n first step will be the separation of the real and imaginary parts by rewriting the identity (4.39) as follows: Re (s) ej2s} = Z [a C (s) + b S (s)] (4.40) n=0 ne ne nono IM (s) ei2is n=la C (s) +b S (s)] (4.41) { ()no not ) fne ne()] (4 ) n=l In Chapter V it will be shown that the theory of discrete Chebyshev approximations provides a means by which the coefficients an and bn can be determined. By this method the coefficients ane and bno are determined such that Re G(s) ej s in a Chebyshev sense at a discrete number of sampling points in the frequency interval of interest. By Chebyshev sense is meant that the maximum deviation of Re G(s) e } from Re{Fl(s) ei2 } is minimum. Similarly the coefficients ano and bne are determined such that Im{G(s) e j2S} approximates Im{ rl(s) ej2ts } in a Chebyshev sense at a discrete number of sampling points in the frequency interval of interest. It will be recalled that the function r1(s) depends on F(s) and rF(s), which in turn depend on the characteristic impedance of the line at its terminals, ZO1 and Z02. The whole synthesis procedure depends on a knowledge of F(s) and Fr(s). It is therefore important to control Z01 and Z02 carefully. This means that the equation (4.38) 1 02 n ( n - = ao +Zb (4.) 2 0 n l no no (.2

40 must be satisfied exactly. It will be shown in Chapter V that the theory of discrete Chebyshev approximation can be used to satisfy some equations exactly, while distributing the approximation error equally over the remaining equations. A special synthesis case arises in the construction of impedance transformers. With the tools developed in this chapter the work of Willis and Sinha (Ref. 13) (see also paragraph 3.3.1) can be extended further. Willis and Sinha considered the case in which p(y) is either of the form N p(y) = Z ane cos negy (4.43) n=On or of the form N p(y) = Z b sin noxy (4.44) n=l Better results can be obtained by considering functions p(y) of the form: N p(y) = Z [a cos neny + bno sin noiy] (4.45) n=0 ne The input reflection coefficient of the line now becomes F (s) ={ [a neCne (s) +bn Sn (s)] e-j23s (4.46) The coefficients ane and bno can be determined to give a high-pass character to r(s) while at the same time minimizing the amount of reflection in the pass band. The theory of discrete Chebyshev approximation

41 provides a very convenient tool to determine the coefficients while at the same time controlling the amount of impedance transformation in the impedance transformer exactly. The amount of impedance transformation is again given by Eq. 4.38. A synthesis example will be given in Chapter VI. This section will be concluded by a short summary of the steps to be taken in the synthesis of a matching section. Given are the internal impedance of a generator as a function of frequency and a load impedance as a function of frequency. It is desired to obtain a match, i.e., maximum power transfer, between generator and load over a band of frequencies which is also given. Step 1. Select the length ~ of the matching section. A suitable choice might be to make the nonuniform line one wavelength long at the lowest frequency of the band of interest. This will determine the frequency variable s. Step 2. Choose Z01 and Z02, the characteristic impedances of the line at its terminals. The magnitudes of the reflection functions r(s) and ro(s) depend on the values chosen for Zol and Z02 It is, therefore, logical to choose Z and Z in such a manner that the maximum values of the magnitudes of the functions F(s) andF F(s) are kept as small as possible. Step 3. Calculate the functions F(s) (Eq. 4.5) and F (s) (Eq. 4.7). These functions are then used to calculate rl(s) (Eq. 4.4). Step 4. Separate the real and imaginary part of rl(s) ej2"s as outlined in the discussion following Eqs. 4.40 and 4.41. Step 5. Determine the coefficients a and b such that the real and imaginary parts of G(s) ej2"s give the best approximation to

42 the corresponding parts of Fl(s) ej2sr. (Eqs. 4.40 and 4.41) This process also involves the selection of the terms that are to be used in the trigonometric expansion (4.12) of the reflection-distribution function p(y). The coefficients an and bn must be determined in such a way that Eq. 4.38, which gives the amount of impedance transformation, is satisfied exactly. The determination of the coefficients a and b essentially concludes n n the synthesis procedure. When these coefficients are known, the reflection-distribution function p(y) is known exactly. The steps necessary to construct a nonuniform line when p(y) is known, are outlined in the next paragraph. 4.4 The Determination of the Characteristic-Impedance Function The characteristic-impedance function Zo(y) can be found immediately by integrating the reflection-distribution function p(y). ~y 1 zo(y) 1 Zo(Y) f p(T) dr = in 0(0) = n. Z(4.47) Therefore: y Z (y) z exp 12 f p(n) dTj (4.48) 0 With p(y) given by a trigonometric expansion (4.12), this becomes Zo(y) = ZO1 exp 2[aOy + Z a sin nvy + b 1- cos nny n o rn=l an sne nity n nt 4.49)

43 characteristic impedance is known at every point in the nonuniform line, as given by (4.48) and (4.49). The physical dimensions of the line are determined directly by Z0(y). In the case of a coaxial structure, for instance, Zo(y) determines the ratio between outer and inner conductor at every point in the line. For a coaxial structure: d. z0(y) d (Y) = exp [~~ ~(] (4.50) where: d is the inside diameter of the outer conductor 0 di is the diameter of the center conductor e is the relative dielectric constant of the dielectric in the line, whose relative permeability is assumed to be unity.

CHAPTER V THE THEORY OF DISCRETE CHEBYSHEV APPROXIMATION 5.1 Introduction In Chapter IV the problem of synthesizing matching sections was reduced to the problem of approximating two given functions, of the variable s, by linear combinations of the functions Cn(s) and Sn(s), which will be called the approximating functions. This is expressed in Eqs. 4.40 and 4.41. In this chapter it will be shown how this problem can be solved by means of the theory of discrete Chebyshev approximation. This theory provides a numerical method by which the coefficients a and b can be determined. It will also be possible to conn n trol the amount of impedance transformation, as given by Eq. 4.38, exactly. The theory of discrete Chebyshev approximation has also proved useful in the synthesis of networks having a prescribed impulse response. This method was recently developed by Ruston (Ref. 10). The treatment in this chapter will follow along lines similar to those used by Stiefel in his recent publication on the theory of discrete Chebyshev approximation (Ref. 12). 5.2 Reduction to an Overdetermined System of Linear Equations A function F(s) is to be approximated by a function f(s), which is a linear combination of n approximating functions fj(s) (1 < j<n): f(s) = xlfl() 2s)s) +... + x f(s) (5.1) where the coefficients xj (1 j i <n) are to be determined such that 44

the maximum value of the magnitude of the approximation error h(s), max Ih(s)l = max JF(s) - f(s)| (5.2) is minimum. The theory of discrete Chebyshev approximation provides a solution to this problem. The functions F(s) and f(s) are sampled at an arbitrarily large number, m, of sampling points, where m > n. Let the m sampling points be sk (1 < k < m), then the function F(s) has to be approximated, at the sampling points, by the function f(s), such that the maximum of the values max Ih(Sk)I = max IF(Sk) - f(sk) (5-3) is minimum. If the approximation of the function F(s) by the function f(s) can be achieved without error at the sampling points, m linear equations Ek (1 < k < m) can be written, one for each sampling point: Ek 1xlfx(sk)+ X2f2(sk) + x nfn(s) - F(sk) = 0 (5.4) n n k k To simplify the notation, the quantities akj and ck will be defined: aJ = fj(sk) (55) Ck = -F(sk) (5.6) For the error h(sk) at the sampling point Sky the notation h will be used.

46 The m equations (5.4) can now be written in the form: Ek: aklXl + ak22 +. + akn +C = 0 (1 <k < m) (5.7) The set of equations (5.4) or (5.7) can be interpreted to represent m planes in n-dimensional Euclidean space. Finding a set of coefficients (x1, x2,..., xn) such that the approximation (5.4) is valid is equivalent to finding the coordinates (xl, x2,..., x) of a point in n-dimensional Eculidean space. If the number (m) of equations equals the dimension (n) of the space, the system (5.4) can, in general, be solved exactly. This means that coefficients (xl, x2,... xn) can be found such that the function f(s) (Eq. 5.1) equals the function F(s) at the m sampling points. In terms of the n-dimensional Euclidean space it means that the m planes (m=n) intersect in a point whose coordinates are (X1, x2,..., Xn). In the case under study, however, the number (m) of sampling points exceeds the number (n) of approximating functions. In that case the system (5.4) becomes overdetermined and it is no longer possible to find a set of coefficients (xl, x2,..., xn) such that all m equations (5.4) are satisfied at once. Or, in other words, when there are m planes in n-dimensional space (m > n), they do, in general, not intersect in one single point whose coordinates are (xl, x2,.., Xn). This case will be studied in the remainder of this chapter. It will be assumed throughout this chapter, that no two planes are parallel. 5.3 Theory of Overdetermined Systems of Linear Equations If a solution exists to the system (5.7), it means that all

m planes in the n-dimensional space intersect in a single point x, whose coordinates are (x1, x2,.., Xn). In general this will not be the case. Consider, therefore, a point P, with coordinates (xi, x,..., ), that does not lie on the plane represented by equation Ek. Substitution of the coordinates of P into the equation Ek, will result in an error, or residue, \, given by: aklXi + ak2X2 + + aknxn + ck = hk (5.8) The approximation problem then becomes that of finding a point T, in n-dimensional space, such that max!hk (1 < k < m) is minimum. This point T will be called the Chebyshev point of the overdetermined system (5-7). In the special case that the normal vectors, nk, to the m planes nk = (akl' ak2'...' akn) (5.9) have unit length, the residue hk represents the distance from point P to the plane Ek, and the approximation problem then becomes the determination of a point T, whose largest distance to any of the m planes is minimum. To find the Chebyshev point, it will be shown that a point P exists, such that the residues hk of a number (n+l) of the m equations (5.7), have equal magnitude Ihkl = jhj. When the selection of these (n+l) planes is such that the magnitude of the error to the remaining equations is less than Ihl, the Chebyshev point has been found. In the following paragraphs it will first be shown how the

48 error to a set of (n+l) equations can be determined. Secondly, it will be shown how a selection of (n+l) planes can be reached to yield the Chebyshev point. A reference will now be defined. A reference is a set of (n+l) equations out of the set (5.7). Without loss of generality, the first (nil) equations of (5.7) can be taken to constitute the first reference. The (n+l) planes represented by these equations have (n+l) normals nk. These (n+l) vectors in n-dimensional space must be dependent, and therefore coefficients k exist such that %1nl + +2n2 + * + ~n+lnn+l = O (5.10) This will be called the characteristic equation of the reference. Because of the assumption that no two planes are parallel, the coefficients Xk are nonzero constants. The characteristic equation (5.10) can also be written as a set of simultaneous linear equations: all1l + a21 2 +... + a(n+l)ln+l = a12X1 + a22 2 +.. * + a(n+l)2n+l = 0 (5.11) alnk1 + a2nk2 +... + a(n+l)n n+l When the coordinates of a point P(x{, xi,.., x) are substituted into the (n+l) equations of the reference, (n+l) equations result: k kl k2X2 +. + aknXn +ck = h (1 < k < n+l) (5.12)

49 By definition, the point P will be called a reference point, when the same condition either sgn hk = sgn X (1 < k < n+l) (5.13) or sgn = - sgn Xk (1 < k < n+l) (5.14) is satisfied by all hk belonging to the reference. The equations Ek of (5.12) are added together after having been multiplied by their corresponding hk's. Because of (5.10) and (5.11), the result of this process is: n+l n+l 2 \ck = 2 Xkhk (5.15) k=l1 K= Because for a reference point either (5.13) or (5.14) is valid, there are now two possibilities. When (5.13) is valid, n+l n+l 2 \kck = IhkI ih k (5.16) k=l k=l When, however, (5.14) is the case, n+l n+l 2 \kck =- I-kI Ih ki (5.17) k=l k=l The significance of the reference point, as defined by (5.13) or (5.14), can be understood using the following argument. First it is observed that, when two points are taken, each lying on different sides of a reference plane, and if one of these points is a reference point,

50 the other is not. This is true because the corresponding error hk has a different sign for the two points, and if condition (5.13) or (5.14) is satisfied for one of the points, it cannot hold for the other. Therefore, the set of reference points forms a continuum, bounded by the planes of the reference set and possibly by infinity. Secondly, because the quantities ck are finite constants, and because the coefficients k are finite nonzero constants, it follows from (5.16) and (5.17) that for all reference points Ihkl is bounded. This proves that the reference points lie inside the volume enclosed by the (n+l) planes of the reference. 5.3.1 Determination of the Center of a Reference. The center of a reference, by definition, is that reference point for which all values Ihk1 are equal. It follows from (5.16) and (5.17) that the reference error, h, equals: 1h = 1 2 +2. n+l n+l1n+ This equation combines Eqs. 5.13 and 5.14. It is easily +erified that, when (5.13) is valid, sgn h = +1, and when (5.14) is valid, sgn h = -1. It also follows, using (5.13) and (5.14), that hk = h sgn \k (5.19) When the reference error (h) has been determined, the coordinates of the reference center can be calculated by solving the following system of linear equations: Ek: kll + ak2x2 +''' + axn + ck = hsgn k (5.20) (1 < k < n+l)

51 In case the normals nk have unit length, the reference center corresponds to the center of the n-dimensional sphere that can be inscribed in the (n+l) reference planes. Of course, the question remains whether a solution exists to the (n+l) equations of (5.20). From (5.11) and (5.15), it follows that the (n+l) equations of (5.20) are linearly dependent, and therefore the existence of the reference center is proved. Because of the assumption that no two planes are parallel, any n equations of the set (5.20) will be independent, and therefore the center of the reference is a unique point. The left-hand side of Eq. 5.16 has a constant value. Therefore, a point P for which one of the errors Ihkf is smaller than Ihl must have at least one other error Ihkl which is larger than jIh. From this it follows immediately that the center of the reference is also the Chebyshev point for the (n+l) planes of the reference. 5.3.2 Overdetermined Systems with Constraints. As was mentioned in Chapter IV, it is necessary to control the amount of impedance transformation, from one end of the matching section to the other, exactly. This implies that Eq. 4.38 must be satisfied exactly. Equation 4.38 also has the general form of Eq. 5.4, and therefore the requirement that it be satisfied exactly means that the equation E0 be satisfied exactly: E a0x + aO2x2 +... + anxn + c = 0 (5.21) The equation (5.21) acts as a constraint on the overdetermined system (5.7). It is now required that a point P (Xl, x2,.., xn) be found such that max fhkt is minimum and such that the point P lies on the plane

52 represented by equation E0,(5.21). Again a reference is defined. It is a selection of n equations from the set (5.7). The first n equations of (5.7) will be chosen to constitute the first reference. The characteristic equation becomes on0 + -1n + 2n2+.. +X n = 0 (5.22) and because the (n-il) normal vectors in n-dimensional space must be dependent (no two are parallel), a set of nonzero coefficients ik (O < k < n) exists such that (5.22) is satisfied. The point P, by definition, is a reference point, when either sgn hk = sgn \ (1 < k < n) (5.23) or sgn hk = - sgn \ (1 < k < n) (5.24) is true for all hk belonging to the reference. The n equations of the reference and equation E0 are added together after having been multiplied by their corresponding \k's. Because of (5.22) and because h0 = 0, the result is: n n Xo ~ + Z ik = Z k k (5.25) k=l k=l Because of (5.23) and (5.24) this can be written as follows: n n ~oo + Z kck = + Z IXkJ Ihkl (5.26) where the positive sign is valid in case Eq. 5.23 holds., and the

53 negative sign when (5.24) is true. The center of a reference is that reference point for which all values Ihkl are equal. The reference error h is found from (5.26): XoC0 + 1X1 + Xc 2 +.. c h = 0 1x 1 2. +nn (5.27) XrJ+ ix21 ~ +. n i and the errors to the individual equations equal: hk = h sgn ik (5.28) The coordinates of the reference center can now be found by solving the following set of simultaneous linear equations: EO' a01X1 + aO2x2 +.. Onxn O (5.29) Ek aklxl + ak2x2 +. + aknxn + ck =hk (1 < k < n) As is apparent from the discussion above, the constraint does not basically alter the procedure for finding the center of a reference. The only difference, as was to be expected, is in the determination of the reference error. Compare Eq. 5.27 with Eq. 5.18. 2.3.3 The Replacement Procedure. After the center of the reference, consisting of the first (n+l) equations of (5.7), has been determined, its coordinates can be substituted into the remaining equations. This substitution gives a set of residues hk (n+2 < k < m). When the magnitude of these residues is smaller than that of the reference error, the reference center is the Chebyshev point for the overdetermined system (5.7).

Suppose, however, that one of the residues, designated by h., has a magnitude larger than Ihl. A new reference must then be selected. This second reference will be formed by replacing one of the equations of the old reference by the equation Ei, whose residue is ho. When the correct equation is replaced, a new reference is generated whose reference error is larger than the old error Ihl. It will be shown next that this is the case when the new set of (n+l) reference planes is selected such that the center (A) of the old reference is a reference point of the new reference. In other words, the (n+l) planes of the new reference must be selected such that the old reference center (A) lies inside the volume enclosed by the new reference planes. That this choice leads to a larger reference error is proved as follows. The new reference consists of n planes of the old reference and the plane Ei. Assume that the plane Er has been replaced. When the coordinates of (A) are substituted into these (n+l) equations, a number n of the residues will have the magnitude Ihj, the remaining one will have the magnitude Ihil, where Ihil > Ihl. When (A) is a reference point of the new reference, Eq. 5.16 or 5.17 holds. Substituting (5.16) and (5.17) into (5.18), one finds the following value for the magnitude of the new reference error h': n+l 1%,j IhiI + IhInl I(kl lX.i + z Iikl k=l; k/r It follows immediately from (5.30), that

55 This completes the proof that the new reference error (h') is larger in magnitude than the old one (h). It now remains to be shown that it is indeed possible to replace one equation by E. in such a way that the point (A) is again a reference point. By replacing the (n+l) planes of the old reference, one at a time, one obtains (n+l) new references, from which a choice must be made. For each of the (n+l) new references a characteristic equation can be written which is of the form (5.10). The characteristic equation for the old reference is: Xn1 + 2n.. +n + +... + n+ln1 = 0 (5.32) The (n+l) normal vectors of the old reference and the normal n. of the plane Ei which is to replace one of these, form a set of (n+2) vectors in n-dimensional space. Therefore, coefficients "k} not all equal to zero, must exist such that -1n+1 +t2nn2 + + C n + A. + + *1n* + nnni = 0 (5.33) By eliminating n (1 < r < n+l) from (5.32) and (5.33), a set of (n+l) characteristic equations is obtained: n n +... +0 n + pK1 -r lnl +[%>-2 - A 2 2 r (3 _j L_2~~~~ (5.34) 3In+l ~r +L -? %n+ln+l + ni where 1< r < n+l. These are the (n+l) characteristic equations of all possible references formed by replacing one plane (Er) of the old reference by

56 the plane Ei. Let primes denote quantities belonging to the new reference. Then (5.34) can be written in the form: Xin ~ %'r + + On + n+ n +... + n +... + ni 0 (5-35) h1 n1 2 2 nr n+l n+2i which is a linear combination of a total of (n+l) vectors. Because the point (A) is a reference point of the old reference, the following condition is satisfied. From (5.19): sgnn = sh sgnhk (1 < k < n+l) (5.36) When the point (A) is also a reference point of the new reference, the following condition must also holdsgnkd = sgn h' sgn hk (5.37) where k assumes the following values k = 1, 2,..., n+l, i; kfr (5.38) For the point (A), hk = hk for k = 1, 2,..., n+l; kfr. Two possibilities now exist. First consider the case in which sgn hi = sgn h and observe that X = +1. The condition (5.37) is fulfilled for all values k of the new reference, as given by (5.38), when sgn k = sgn X9. This condition will be fulfilled when that plane Er is replaced for which r the ratio -* is minimum. This follows from the fact that, when r hr P then:

57 h _ r > O (5.40) Secondly, when sgn h. = - sgn h, condition (5.37) will be satisfied when sgn S = - sgn \. This can be achieved by replacing that plane Or E for which the ratio -is maximum. In that case, r r'Lr'k > (k = 1, 2,..., n+l; kfr) (5.41) r Xk and hi k $ < 0 (5.42) This completes the replacement rule, which can be summarized as follows: when sgn h = sgn h, replace the plane for which - is minimum (5.43) when sgn h. = -sgn h, replace the plane for which -k is maximum After replacement, a new reference is obtained whose center can be determined with the procedure discussed in paragraph 5.3.1. The coordinates of the new reference center are then substituted into the remaining equations of the overdetermined system (5.7). When any of the resulting errors are larger than the reference error Ih' j, the replacement process must be undertaken a second time.

58 Every time the replacement process takes place, the new reference error is larger than the previous one. After a finite number of steps, the replacement process must terminate, because there are only a finite number (m) of equations. Also, the same reference will never be arrived at twice in this process, because the reference error Ihi increases monotonically. The center of the last reference is the Chebyshev point for the overdetermined system (5.7). When the overdetermined system is subject to a constraint, the replacement process remains unaltered, except for the fact that the constraint equation E0 is not eligible for replacement. 34 Examle. To conclude Chapter V, an example of an overdetermined system will be given. Consider the overdetermined system E: x + 2 2 + 5 2' 3 x + x2 + +2 0 (5.44) E 2 x1 + 3 x2 + 7 O0 subject to the constraint that the equation EO E0 X1 + x2 - 3 = (5.45) be satisfied exactly. The equations Ek represent planes (lines) in two-dimensional space. The normals to these lines are: nO = 1, 1 nl = 1, 2 (5.46) n2 = 3, 1 n3 = 2, 3

59 Let the first reference consist of the equations E1 and E2. Then the characteristic equation can be written according to (5.22): ono + lnl +2n 2 = 0 (5.47) The coefficients k are determined by solving the following set of simultaneous linear equations, corresponding to (5.47): X0 + k1 + 3x2 = 0 (5.48) + 2+ 1 + 2 = 0 The set of %'s satisfying these equations is: x =-5; X 2; X le 0 1=; k2 = 1. With these values, the reference error h can be calculated using (5.27). h _. -5)(-3) + (2)(5) + 9()(2) 9 (5.49) 121 + il The errors to the reference equations are hk = h sgn 1K. Substituting these errors into the set of reference equations gives: E0 xl1 + x2 - 3 0= E1: xl + 2 2+ 5 = 9 (550o) E2' 3 x + x2 + 2 = 9 2 1 2 The center of the reference is found by solving this set of equations. One finds that: xl = 2; x2 = 1 (5.51)

60 It is easily verified that these valus for xl and x2 satisfy all three equations (5.50). The error of the remaining equation E3 is next determined. Substituting the values (5.51) into E3, one finds chat the error h3 equals 14, which is obviously larger than the reference error. A new reference must now be chosen, using the replacement process. First, coefficients,i are determined such that 0on0 + l1n1+ i2n2 + n3 = 0 (5.52) One of these coefficients. can be chosen arbitrarily. Let 1O 0. Then the following system of linear equations has to be solved: _1 + 3 ~2 + 2 = 0 (5-53) 2 1 L+ PL2 + 3 = 0 The following values of p satisfy (5.53): =1 7 =L I(5.54) The ratio k can now be determined for the two planes of the old reference. A1 7; ~12~ 1 ~(5 55) 1 t10' (2 5 Because h = 9 (5.49) and h = 14, sgn h = sgn h. According to rule (5.43), that plane has to be replaced for which is minimum. From (5.55) it is found that equation E1 has to be replaced. The new reference then consists of equations E2 and E3, subject to the condition EB.

61 The characteristic equation for the new reference is written in the form of two linear equations as follows: X0 + 3 X2 + 2 3 ~ 0 (5.56) N0 + X2 + 3 3 = 0 The following values for ik satisfy these equations: X0 = - 71; 2 2 (5 57) The new reference error h is calculated: h = (7)(-3) + (1)(2) + (2)(7) 37 (5.58) J=l + 121 3 The new reference center is found by solving the following set of equations: E: x1 + x -3 = 0 2' 3 x + x2 + 2 = 123 (559) E3 2 xl + 3 x2 + 7 = 123 and the coordinates of the reference center are: 11 2 Xl = = -3 (5.60) As a final check, these coordinates are substituted into equation El, which gives hI = 7-. This is smaller than the reference error, and the Chebyshev point for the system therefore has the coordinates (5.60).

CHAPTER VI EXAMPLES 6.1 Introduction In this chapter some examples will be given of the synthesis of nonuniform lines, using the tools developed in Chapters IV and V. In Section 6.2 the theory of discrete Chebyshev approximation will be applied to the synthesis of impedance transformers. In Section 6.3 a matching section will be synthesized to provide a match between a generator, with an internal impedance of 50 ohms, and a 100-ohm load with stray capacitance. When a given function is approximated by a linear combination of approximating functions, using the theory of discrete Chebyshev approximation, the maximum error at the sampling points will be minimum. One has essentially no control over the behavior of the function between the sampling points. However, by choosing a sufficiently large number of sampling points, spaced closely together, one can be confident that the error between sampling points will not exceed the error at the points by any appreciable amount, so that, for all practical purposes, a true Chebyshev approximation is obtained. In the examples given in this chapter the sampling points will be chosen at integral values of the independent variable 4s. The approximating functions Cn(s) and Sn(s) either have the value zero at these points, or reach an extreme value in the close vicinity of these sampling points (see Figs. 4.2 and 4.3). Because of the smooth behavior of the functions Cn(s) and Sn(s), good results are obtained using these sampling points. The functions Cn(s), evaluated at integral values of the variable 4s, are given in Table I. The functions Sn(s) are given in Table 62

4s CO(s) C1(S) (2(S) C3() 14(s) C5(S) C6(s) C7(S) C8(s) 09(s) C10(s) C11(S) C (13(S) C15(s 0 1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.63662 0.5 -0.21221 0 -0.04244 0 -0.01819 0 -0.01010 0 -o.oo644 0 -0.00445 0 -0.00326 0 2 0 0.42441 -0.5 -0.25456 0 -o.o6063 0 -0.02829 0 -0.01654 0 -0.01089 0 -0.00771 0 -0.00576 3 -0.21221 0 -0.38197 -0.5 0.27284 0 0.07073 0 0.03473 0 0.02099 0 o0.1415 0 0.01021 0 4 0 -0.16976 0 -0.36378 0.5 0.28294 0 0.07717 0 0.03918 0 0.02425 0 0.01665 0 0.01218 5 0.12732 0 0.15157 0 0.35368 0.5 -0.28938 0 -0.08162 0 -0.04244 0 -0.02675 0 -0.01862 0 6 0 0.10913 0 0.14147 0 0.34724 -0.5 -0.29383 0 -0.08488 0 -o.o4494 0 -0.02872 0 -0.02021 7 -o.o9094 0 -0.09903 0 -0.13503 0 -0.34279 -0.5 0.29709 0 0.08738 o o.o4691 0 0.03031 0 8 0 -0.08084 0 -0.09259 0 -0.13058 0 -0.33953 0.5 0.29959 0 0.08935 0 0.04850 0 0.03163 9 0.07074 o 0.07440 0 0.08814 o 0.12732 0 0.33703 0.5 -0.30156 0 -o.o9094 0 -o.o4982 0 10 0 0.06430 0 o.o6995 0 0.08488 0 0.12482 0 0.33506 -0.5 -0.30315 0 -0.09226 0 -0.05093 11 -0.05786 0 -0.05985 0 -o.o6669 0 -0.08238 0 -0.12285 0 -0.33347 -0.5 0.30447 0 0.09337 0 12 0 -0.05341 0 -0.05659 0 -o.o6419 0 -0.08041 0 -0.12126 0 -0.33215 0.5 0.30558 0 o.o9431 13 0.04896 o 0.05015 0 0.05409 0 0.06222 0 0.07882 0 0.11994 o 0.33104 0.5 -0.30652 0 14 0 0.04570 0 0.04765 0 0.05212 0 0.06063 0 0.07750 0 0.11883 0 0.33010 -0.5 -0.30733 15 -0.04244 0 -0.04320 0 -0.04568 0 -0.05053 0 -0.05931 0 -0.07639 0 -0.11789 0 -0.32929 -0.5 TABLE I. The functions C (s) and Cn(s) for integral values of (4s). ne no

4s S1(s) S2(S) 3(s) )' 6 S S(s) S9(s) S0(S) S1(S) S2(S) S3(S) S14(s) S15(s) 0 0.63662 0 0.21221 0 0.12732 0 o.09094 0 0.07074 0 0.05786 0 o.04896 0 0.04244 1 0.5 0.42441 0 0.16976 0 0.10913 0 o.o8o84 0 0.06430 0 0.05341 0 0.04570 0 2 0.21221 0.5 -0.38197 0 -0.15157 0 -0.09903 0 -0.07440 0 -0.05985 0 -0.05015 0 -0.04320 3 0 0.25465 -0.5 -0.36378' o -0.14147 0 -0.09259 0 -o.o6995 0 -0.05659 0 -0.04765 0 4 -0.04244 0 -0.27284 -0.5 0.35368 0 0.13503 0 o.o88i4 0 0.06669 0 0.05409 0 o.o4568 5 0 -0.06063 0 -0.28294 0.5 0.34724 0 0.13058 0 0.08488 0 0.06419 0 0.05212 0 6 0.01819 0 0.07073 0 0.28938 0.5 -0.34279 0 -0.12732 0 -0.08238 0 -0.06222 0 -0.05053 7 0 0.02829 0 0.07717 0 0.29383 -0.5 -0.33953 0 -0.12482 0 -0.08041 0 -0.06063 0 0>\ 8 -0.01010 0 -0.03473 0 -0.08162 0 -0.29709 -0.5 0.33703 0 0.12285 0 0.07882 0 0.05931 9 0 -0.01654 0 -0.03918 0 -0o08488 0 -0.29959 0.5 0.33506 0 0.12126 0 0.07750 0 10 0.00644 0 0.02099 0 0.04244 0 0.08738 0 0.30156 0.5 -0.33347 0 -0.11994 0 -0.07639 11 0 0.01089 0 0.02425 0 o.o4494 o 0.08935 0 0.30315 -0.5 -0.33215 0 -0.11883 0 12 -0.00445 0 -0.01415 0 -0.02675 0 -o.o4691 0 -o.o0994 0 -0.30447 -0.5 0.33104 0 0.11789 13 0 -0.00771 0 -0.01665 0 -0.02872 0 -0.04850 0 -0.09226 0 -0.30558 0.5 0.33010 0 14 0.00326 0 0.01021 0 0.01862 0 0.03031 0 0.04982 0 0.09337 0 0.30652 0.5 -0.32929 15 0 0.00576 0 0.01218 0 0.02021 0 0.03163 0 0.05093 0 0.09431 0 0.30733 -0.5 TABLE II. The functions S nes) ad S no(s) for integral values of (4s).

65 II. As can be seen from these tables and also from Figs. 4.2 and 4.3, the functions Cn(s) and Sn(s) consist of a main lobe around the point 4s = n and smaller side lobes. The farther the side lobe from the main lobe, the smaller its amplitude. When a large number of functions Cn(s) and S (s) are used in the approximation process, the amount of computation could be reduced if some means could be found to let the amplitude of the side lobes approach zero more rapidly. Danielson and Lanczos have developed a transformation by which the set of functions Cn(s) and Sn(s) is transformed into a new set of functions whose side lobes approach zero more rapidly than those of the original functions. Their method can be modified to apply to the present problem. This process is treated in the appendix. 6.2 Snthesis of Impedance Transformers The first two problems to be considered will be the synthesis of two impedance transformers, one 0.75 X long at the lowest frequency of the pass band, the other 1 X long. Results are obtained that represent an improvement over those obtained by Willis and Sinha (paragraph 3.3.1), partly because of the more general synthesis formula (Eq. 4.45 and 4.46), and partly because of the use of the theory of discrete Chebyshev approximation. This theory, developed in Chapter V, provides a very powerful tool, far superior to the trial-and-error method used by Willis and Sinha. A large number of terms in the expansion (4.45) can be handled conveniently, while, at the same time, the amount of impedance transformation (4.38) is controlled directly. 6.2.1 0.75 A Transformer. To determine which terms should be used in the trigonometric expansion (4.12) of the reflection

66 distribution function p(y), a qualitative argument can be used, involving Parseval's theorem and the equation giving the amount of impedance transformation (4.35). Parseval's theorem, applied to the present problem, states that: co 2 1 2 2 f lr(s) 2 ds - j p(y) 2 dy (6.1) -When the am ount of impedance 0 When the amount of impedance transformation is given, the following integral (4.35) is determined: 1 z f p(y)' dy = ln 02 (6.2) 0'2 ZO1 0 01 where p(y), of course, is a real function. One can state that qualitatively the amount of impedance transformation, in first approximation, determines the value of the integral (6.1). If r(s) has a high-pass character, it immediately follows from (6.1) that the area of the main lobe must increase when the reflections in the pass band decrease. To increase the value of TF(s)j outside the pass band, functions Cne(s) and Sn (s) can be selected such that their main lobes fall outside the pass band. A suitable choice for a 0.75 X transformer would be the functions CO(s), C2(s), and Sl(s). The reflection-distribution function is then expanded as follows: p(y) = a + a2 cos 2gy + b sin iy (6.3) It follows from (4.46) that r(s) ej2s = aOCO(S) + a2 C2(s) + bI Sl(S) (6.4) The input reflection r(s) should approximate the value zero for all values of the variable s equal to and greater than 0.75.

67 Using Tables I and II, one finds the following equations (Ej) at the sampling points, where the subscripts correspond to the value of 4s at the sampling point. The equation E0 is the equation corresponding to Eq. 4.38, determined by the amount of impedance transformation. This equation has to be satisfied exactly and the error has to be distributed over the remaining equations. The amount of impedance transformation will be choZ2 2 sen equal to 02 = e so that: 01 z02 1In 02 (6.5) A direct comparison is then possible with impedance transformers given in the literature and discussed in paragraph 3.3.1. Ed: 1.0 a0 +0.6366 bl -1.0 0 o E3: -0.2112 a0 -0.3820 a2 0 E4: -0.0424 b 1 0 E5' 0.1273 a0 +0.1516 a2 0 E6: 0.0182 b1 = 0 E7 -0.0909 a0 -0.0990 a2 0 8 -0.0101 b1 = 0 E9: 0.0707 a0 +0.0744 a2 0 E10: 0.0064 b = 0 E11: -0.0579 a0 -0.0598 a2 = 0

E12: -0.0044 bl = 0 E13: 0.0490 a0 +0.0502 a = 0 El4: 0.0033 b1 = 0 E15: -0.0424 a0 -0.0432 a2 = 0 (6.6) Obviously the set of equations (6.6) cannot be satisfied simultaneously. A reference is chosen consisting of equations EO, E3, E4, and E5. Four equations are taken because the space is three-dimensional. Next, a set of Xj's are calculated such that 0n0 + k3n3 + X4 4+ nsn = 5 (6.7) When these Xk's are known, the Chebyshev error for the chosen reference can be determined. Equation 6.7 can be written in the form of a set of simultaneous linear equations as follows: 1.0 k0 -0.2122 X3 +0.1273 X5 = 0 -0.3820 X3 +0.1516 k5 = 0 (6.8) 0.6366 Xo -0.0424?4 = O The set of X. satisfying these equations is: Mo = 1; X3 = -9.2036; A = 15.0005; = -23-1935 0 3 4 5 The Chebyshev error is determined next (5.27): XoCo + X3c3 + 4c4 + X5c5 h= 00.... 3344... 5 5 = -0.0211 (6.9) iX31 + 1X41 + s1X51

69 The error of the jth equation, h., equals hj = h sgn X = -0.0211 sgn kX (6.10) This error is applied to the equations of the reference, which leads to a set of equations that can now be solved. EO: 1.0 ao +0.6366 bl -1.0 = 0 E3: -0.2112 a0 -0.3820 a2 0.0211 3: E4: -o.0424 b = -0.0211 (6.11) E5: 0.1273 a0 +0.1516 a = 0.0211 The following set of coefficients satisfies these four equations: a0 = 0.6835; a2 = -04350; b1 = 0.4972 (6.12) These values for the coefficients are substituted in the equations E6 through E15 to determine the error at the other sampling points. One finds the following values for these errors: h6 = 0.0090 h7 = -0.0191 h8 = -0.0050 h9 = 0.0160 h10 = 0.0032 h11 = -0.0135 h12 = -0.0022 h13 = 0.0117 h14 = 0.0016 h15 = -0.0102 (6.13) It is easily verified that all these errors are smaller than the reference error (6.9). This, therefore, concludes the determination of the coefficients aO, a2, and bL. The reflection-distribution function for this impedance transformer is equal to

70 p(y) = 0.6835 - 0.4350 cos 2ty + 0.4972 sin iy (6.14) The reflection coefficient of the line is plotted in Fig. 6.1. As was found above (Eq. 6.9), the maximum reflection in the pass band is 0.0211. This value can be compared with the results obtained by Willis 1I..8.4.2 G.5 1 1.0 1.5 2.05.02 o0.5 1.0.5 2.0 2.5 - S Fig. 6.1 Reflection pattern for a 0.75 X transformer. and Sinha and by Klopfenstein. The reflection in the pass band in the example of Willis and Sinha, for an impedance transformer of the same length, equals 0.031 (see Fig. 3.1). The present design compares favorably with this and is almost as good as the Chebyshev taper which gives reflections in the pass band equal to 0.018. The Chebyshev taper has the disadvantage of having discrete impedance steps at the two ends of the line. The line designed in this paragraph is smooth everywhere. From the reflection-distribution function the characteristic impedance everywhere in the line can be calculated using Eg. 4.48. The reflection-distribution function p(y) is plotted in Fig. 6.2. The

71 pl(y) 0 2.5.5.75 1.0 -o y Fig. 6.2 Reflection-distribution function for a 0.75 X transformer. Zo(y) zo' 16 4 2 - 2.5 5 75 1.0 Fig. 6.3 Characteristic impedance function for a 0.75 x transformer.

72 characteristic-impedance function for the whole line is plotted in Fig. 6.3. 6.2.2 1 X Transformer. The next example of synthesis will be an impedance transformer which has a length of one wavelength at the lowest frequency of the pass band. In this example the functions Co(s), C2(s), C4(s), and C6(s) will be chosen to synthesize a reflection coefficient r with high-pass character. This implies that the reflectiondistribution function p(y) is of the form: p(y) = a0 + a2 cos 2gy + a4 cos 4ny + a6 cos 6,ty (6.15) The corresponding input reflection coefficient can be written according to Eq. 4.46: F(s) ejs = a oCo(s) + a2C2(s) + a4C4(S) + a6C6(S) (6.16) The only term in the expansion of p(y) that contributes to impedance transformation is the constant term (ao). The restriction placed on the overdetermined system by the amount of impedance transformation now becomes a trivial one. In this example, therefore, an overdetermined system will be considered that is not subject to constraints. The coefficient a0 is directly determined by the amount of impedance transformation [Eq. 4.38]: 1 _o2 a0 1 = n 02 (6.17) o 2 Z1 Because only cosine terms are used in the expansion (6.15), the function r(s) will be zero for even integral values of 1s larger than 6. The corresponding equations, therefore, do not appear among the ones

listed below. The following set of equations can now be written, using Table I. E4: 0.5000 a4 = 0 E5. 0.1516 a2 +0.3537 a4 -0.2894 a6 +0.1273 = 0 E6: -0.5000 a6 = 0 E7: -0.0990 a2 -0.1350 a4 -0.3428 a6 -0.0909 = 0 (6.18) E9: 0.0744 a2 +0.0881 a4 +0.1273 a6 +0.0707 = 0 E1: -0.0598 a2 -0.0667 a4 24 -0. 0579 = 0 E13: 0.0502 a2 +0.0541 a4 +0.0622 a6 +0.0490 = 0 El5: -0.0432 a2 -0.0457 a4 -0.0505 a6 -0.0424 = 0 A reference is chosen out of this set, consisting of equations E4, E5, E6, and E7. Four equations are taken because the space is three-dimensional. A set of Xj's for this reference is calculated to satisfy: X4n4 + X5n5 + 6n6 + X7n7 = O (6.19) This can be written in the form of a set of simultaneous linear equations as follows: 0.1516 5 -0.0990 X7 = 0 0.5000 k4 +0.3537 k5 -0.1350 k7 = 0 (6.20) -0.2894 k5 -0.5000 k6 -0.3428 7 = O The following set of values satisfies these equations:

74 X = -0.1921 k5 = 0.6534 k6 = -1.0637 7 1.0 4 ) 6 (6.21) The reference error is next determined (5.18): X4c4 + 5 C5 + k6C6 + 7 c7 h = -0.0027 (6.22) 1X41 + 1x51 + 1x6i + 1x71 The error to the jth equation (hj), equals h. = h sgn Xj. This error is added to the equations of the reference, which leads to a set of equations that can now be solved. E4 0.5000 a4 = 0.0027 E5 0.1516 a2 +0.3537 a4 -0.2894 a6 +0.1273 = -0.0027 (6.23) E6~ -0.5000 a6 = 0.0027 E7: -0.0990 a2 -0.1350 a4 -0.3428 a6 -0.0909 = -0.0027 The following set of coefficients satisfies these equations: a0 = 1.0; a2 = -0.8598; a = 0.0053; a6 = -0.0053 (6.24) These values for the coefficients can be substituted in the remaining equations to determine the error at the other sampling points. One finds: h 0.0066; h = -0.0063; h3 = 0.0058; h = -0.0053 (6.25) All these errors happen to be larger than the reference error. The synthesis problem would now be solved if one were interested only in obtaining an impedance transformer over a frequency range corresponding

75 to a range of values for s between one and two. In the problem considered in Section 6.3 such an impedance transformation will be needed, and the coefficients given in Eq. 6.24 will be used. In the present problem, however, a high-pass characteristic is desired. The replacement process will then be necessary to determine which of the equations of the reference (6.23) must be replaced. The largest of the errors (6.25) is h9. The equation Eg will therefore be used to replace one of the equations of the old reference. Replacement by the equation possessing the largest error does not guarantee success in the next attempt, but it is the most appropriate choice under the circumstances. The replacement process starts by determining the coefficients.j that can be found by solving the equation: [44n4 + 5n5 + 6n6 + 7n7 + n (6.26) This equation can be written as a set of simultaneous linear equations. 0.1516 45 -0.0990 p.7 +0.0744 = o 0.5000 p.4 +0.3537 p5 -0.1350 p7 +0.0881 = 0 (6.27) -0.2894 5 -0.5000 1p6 -0.3428 j17 +0.1273 = 0 The following set of 1a's is one of the infinitely many that satisfy Eq. 6.27 p4 = 0.0116; p15 = 0; p16 = -0.2604; p7 = 0.7513 (6.28) Using these values, the ratio can be determined:

76 -0-.0603; = 0; =.2450; (6.29) k4 60.7513.29x Because h < 0 (6.22) and h9 > 0 (6.25), that equation from the old reference must be replaced, for which the ratio -~ is maximum, according to rule (5.43). Therefore equation 7 has to be replaced so that the new reference consists of E4, E5, E6, and Eg9. Again coefficients Xj are determined such that X n Xn + sn5+ -n6 + n = 0 (6.30) This can be written as a set of simultaneous linear equations as follows: 0.1516 k5 +0.0744 k9 = 0 0.5000 X4 +0.3537 k5 +0.0881 k9 = 0 (6.31) -0.2894 k5 -0.5000 k6 +0.1273 k9 = O The following Xj's satisfy these equations k4 = 0.1709; k5 = -0.4909; X6 = 0.5387; X9 = 1.0 (6.32) These values are used to calculate the new reference error: x4c4 + X5C5+ X6C6 + X9c9 hhc4 = \5c5 + A6C6 9 9= 0.0037 (6.33) 1x41 + j151 + 1X61 + 1X91 The set of reference equations can now be solved by introducing the error into the equations:

77 E4 ~ 0.5000 a4 = 0.0037 E': 0.1516 a2 +0.3537 a4 -0.2894 a6 +0.1273 = -0.0037 (6.34) E6: -0.5000 a6 = 0.0037 Eg: 0.0744 a2 +0.0881 a4 +0.1273 a6 +0.0707 = 0.0037 The solution of this set yields the following values for the coefficients: a2 = -0.8964; a4 = 0.0075; a6 = -0.0075 (6.35) These values are substituted into the remaining equations to determine the error at the other sampling points. One finds: h7 = -0.0006; h11 = -00041; l3 = 0.0039; h15 - -0.0037 (6.36) It can be seen that hll exceeds the reference error, and therefore the replacement process has to be undertaken a second time. Again coefficients.j. are determined such that 4L4n4 + 5 n + 5 n 6+ 9n9 + ll = 0 (6-37) Written as a set of simultaneous linear equations this becomes: 0.1516 ~5 +0.0744 u9 -0.0598 = 0 0.5000 44 +0.3537,5 +0.0881 p9 -0.0667 = 0 (6.38) -0.2894 45 -0.5000 16 +0.1273 49 -0.0824 = 0 The following coefficients pj satisfy these equations: 44 = -0.0084; l5 = 0; 46 = 0.0401; 4g = 0.8044 (6.39)

78 LLj The ratios J- are then determined using (6.39) and (6.32): 4= -0.0492; = 0. 0744; - = 0.8044 (6.40) -o.o49;6 90744 Because h > 0 (6.33) and hll < 0 (6.36), that plane of the reference 1j must be replaced for which the ratio. is maximum. Therefore equation 9 is replaced by equation 11, and the new reference consists of equations E4, E5B E6, and El,. First the coefficients Xj are determined again such that 4 4 (5 5 6. )1111 This leads to the following set of equations: 0.1516 X5 -0.0598 kli = 0 0.5000 AI +0.3537 X5 -0.0667 XA 0 (6.42) -0.2894 k5 -0.5000 X6 -0.0834 Xll = 0 with the solution: = -0.1459; 0.3949; k6 = -0.3933; 011 = 1.0 (6.43) Using these values, the reference error is determined: X4c4 +X..c5 +X6C6 +xllc11 Ih + 55 6 1111 = -0.0039 (6.44) IX41 + IX51 + IX61 + 1Xll1 The reference set is now written as follows, including the error:

79 E4: 0.5000 a4 = 0.0039 E: 0.1516 a2 +0.3537 a4 -0.2893 a6 +0.1273 = -0.0039 (6.45) E6: -0.5000 a6 = 0.0039 E11: -0.0598 a2 -0.0667 a4 -0.0824 a6 -0.0579 = -0.0039 The solution is: a = 1.0; a= -0.8991; a= 0.0078; a= -0.0078 (6.46) These values are then substituted in the remaining equations to determine the error at the different sampling points: h7 = -0.0003; h = 0.0035; hL3 = 0.0038; h15= -0.0036 (6.47) All these individual errors are smaller than the reference error and therefore the process is completed. The reflection-distribution function for the impedance transformer is now equal to: p(y) = 1.0 - 0.8991 cos 2gy + 0.0078 cos 4ty - 0.0078 cos 6iy (6.48) The reflection pattern for this line is plotted in Fig. 6.4. According to Eq. 6.44 the maximum reflection in the pass band equals 0.0039. This value can be compared with the results obtained by Willis and Sinha and by Klopfenstein (see paragraph 3.3.1). For a 1 X transformer Willis and Sinha find a reflection in the pass band of 0.0056, and the reflections for the Dolph-Chebyshev taper are 0.0037. It is seen that the line synthesized above is almost as good as the Dolph-Chebyshev line, while it does not have the disadvantage of the discrete impedance jumps

80.3 IrI.2.05 I.0.0039 0.5 1.0 1.5 2.0 2.5 S Fig. 6.4 Reflection pattern for a 1 X transformer. at the ends of the line. The reflection-distribution function (Eq. 6.48) for the line is plotted in Fig. 6.5. The characteristic-impedance function Eq. 4.48 is plotted in Fig. 6.6. 6.3 Synthesis of a Matching Section As a third example a matching section will be synthesized that matches a generator, with real internal impedance (50 ohms), to a complex load. The load is a 100-ohm resistor with 1 pf stray parallel capacitance. A uniform line can be used to connect the generator to the matching section, as indicated in Fig. 6.7. The matching section will provide a match over a frequency range extending from 500 to 1000 Mc, and the length of the mnatching section will be 60 cm, which

81 p(y) 0.25.5.75 1.0 - y Fig. 6.5 Reflection distribution function for a 1 X transformer. 8 Zo(y) zol 6 4 0O 0.25.5.75 1.0 - y Fig. 6.6 Characteristic-impedance function for a 1 X transformer.

82 corresponds to one wavelength at 500 Mc, the lowest frequency of the pass band. When the load Z2, as shown in Fig. 6.7, is connected directly to the terminals of the uniform line, omitting the matching section, it will cause a reflection of magnitude Z - 50 Ifrlj =l2 + < 0.39 (6.49) z2+50 oxl ('~') zo= 5on Ipf,Ioon GENERATOR UNIFORM LINE LOAD MATCHING SECTION Fig. 6.7 Circuit diagram for the matching example. which corresponds to a voltage standing wave ratio (VSw4R) of 2.26 in the 50-ohm line. If, instead of the matching section, a 1 X impedance transformer is used, best results are obtained when the impedance transformer transforms from 50 to 84.7 ohms. (This fact will be discussed further on in this section.) Using this optimum transformer, the VSWR in the 50-ohm line will equal 1.81. In this section two matching sections will be synthesized, giving standing wave ratios equal to 1.07 and 1.10, respectively. The first step in the synthesis procedure is to choose the length of the matching section. In this example the matching section

83 will be one wavelength long at the lowest frequency, 500 Mc. The matching section will therefore by 60 cm long. The frequency variable s is then determined. s varies linearly with frequency, and s = 1 for 500 Mc, and s = 2 for 1000 Mc (Eq. 3.21). The second step is to choose the value of the characteristicimpedance function at the two terminals of the matching section, Z01 and Z02 Zo1 will be chosen equal to 50 ohms, so that the section matches the generator. Z02 will be chosen such that the maximum value reached by ir2(s)|, as defined by Eq. 4.6, is minimum. r2(s) can be written as follows: ~1 1 1 - 1 -OC 0.01 - j 0.0031 s Z02 2 Z 02 R 02 (S) -. 1 1s1 I 22 02 - + z 1Z R jWC iz + 0.01 + j 0.0031 s Z02 2 Z02 R 02 (6.50) IF2(s)I reaches its maximum value at the highest frequency of the pass band, when s = 2. Setting s = 2 in Eq. 6.50, it can be shown that the value of 1F2(2)1 reaches a minimum when Z02 = 84.7 ohms (6.51) This value of Z02 will be used in the example. 02 The functions r(s) and rl(s), as defined in Eqs. 4.5 and 4.7, can now be calculated. Obviously r(s) = O, and rI(s) is given by (compare Eqs. 6.50 and 6.51): 0. 0018 - j 0.0031 s -j4ns F (s) = j.0~'8 + j 0.0031 s e (6.52) rl(s)' as defined in Eq. 4.4, becomes (4.10):

84 Fl(s) = - 7(s) (6.53) In the synthesis procedure, the real and imaginary parts of Fl(s) eJ2rs are needed. They can be calculated using (6.52) and (6.53) and are tabulated below for integral values of 4s, which are the sampling points, in the frequency range of interest. r(i) e2s =Oo0.0018 - 0.0031 s e-2s (6 rl (s) e2+ e (6.54) l 0 o.0218 + j 0.0031s 4s Re{I1(S) ej2gs} Im{rT(s) ej2S}l 4 -0.0610 0.1528 5 0.1888 0.0490 (655) 6 0.0347 -0.2236 7 -0.2567 -0.0183 8 0.0 0.2881 A function G(s) is to be found that approximates rl(s), where G(s) is the Fourier transform of a real function p(y), which is zero outside the interval (0,1). First, the real part of G(s) ej2ns will be synthesized. According to Eq. 4.40: Re {G(s) ej2} = [ neCne(s) + bnoSno(s)] (6.56) It can be seen from the table (6.55) that Re{ rL(s) ej2ts } is small for even values of 4s. Because of this, the functions S3(s), S5(s), S7(s), and S9(s), will be used for the approximation process. There are five sampling points, and therefore the four chosen functions lead to an overdetermined system. There is also the Eq. 4.38 which controls the amount of impedance transforation along the line. This equation has to be satisfied exactly. When this overdetermined

85 system, with constraint, is solved, the error will prove to be too large. Therefore more approximating functions, for instance C2(s), C4(s), etc., are needed, leading to an overdetermined system of higher dimension. To avoid the tremendous amount of computation involved in solving this system, a slightly different approach will be taken in this example. First, the coefficients b3, b5, b7, and bg will be determined, without regard to impedance transformation. The error in the impedance transformation will then be corrected using the functions C0(s), C2(s), C4(s), and C6(s), without essentially disturbing the approximation. The approximation process is started by writing the equations for the sampling points, using Table II and (6.55). E4: -0.2728 b3 +0.3537 b5 +0.1350 b7 +0.0881 b9 +0.0610 = 0 E5 o 0.5000 b5 -0.1888 = o 5 5 E6: 0.0707 b3 +0.2894 b5 -0.3428 b7 -0.1273 b9 -0.0347 = 0 (6.57) E7: -0.5000 b7 +0.2567 = 0 E8: -0.0347 b3 -0.0816 b5 -0.29'71 b7 +0.3370 b9 = 0 To determine the Chebyshev error for this system, coefficients Xj must be determined such that X nq + Xi + X,-n + Xn+ Xnn 0 (6.58) 4 4 55 5 6 6 T7 8 - This leads to the following system of simultaneous equations. -0.2728 X4 +0.0707 6 -0.0347 8 = 0 0.3537 X4 +0.5000 X5 +0.2894 X6 -0.0816 k = 0

86 0.1350 h4 -o.303428 X6 -0.5000 X7 -0.2971 k8 = 0 0.0881 x4 -0.1273 A6 +0.3370 k8 = 0 (6.59) This set is satisfied by the following set of values for X.: 4 = 0.6812; X5 =-2.1236; X6 = 3.1187; 7 =-2.5483; k8 = 1.0 (6.60) Using these values for Xj, the Chebyshev error for the system (6.57) can be determined. %4c4 + X5C5 + X6C6 + X7C7 + X8C8 h. 5 668 = -0.0338 (6.61) 1x41 + IX51 + x6 + 7+ 1X81 The error h. of equation Ej is then determined by hj = h sgn Xk, so that the following set of equations results, that can now be solved. E4: -0.2728 b3 +0.3537 b5 +0.1350 b7 +0.0881 b9 +0.0610 = -0.0338 E5: 0.5000 b5 -0.1888 = 0.0338 E6: 0.0707 b3 +0.2894 b5 -0.3428 b7 -0.1273 bg -0.0347 = -0.0338 E7: -0.5000 b7 +0.2567 = 0.0338 E8: -0.0347 b3 -0.0816 b5 -0.2971 b7 +0.3370 b9 = -0.0338 (6.62) The coefficients that satisfy these equations are: b3 = 1.3181; b5 = 0.4451; b7 = 0.4458; b9 = 0.5363 (6.63) ehe amount of impedance transformation was not controlled in the above procedure. The amount of impedance transformation is given by 13q. 4.38,

87 which becomes: 0.2122 b3 + 0.1273 b5 + 0.0909 b7 + 0.0707 b9 = 0.4149 (6.64) The correct amount of impedance transformation is given by 1 Z0( 1 84.765) i:n Z0 2 in 407 =0.2653 (6.65) 2 Z01 2 50 A correction to Eq. 6.64, equal to -0.1496, is needed to obtain the correct amount of impedance transformation in the line. The first approximation to the impedance transformer synthesized in paragraph 6.2.2 is ideally suited for the purpose of providing the necessary correction terms. One will recall that the coefficients for this transformer, as written in Eq. 6.24, provide minimum reflection inside the frequency band 1 < s < 2, while the reflections outside this band are larger. By multiplying the coefficients, given in (6.24), by the factor -0.1496, the following coefficients are obtained that will give the correct amount of correction to Eq. 6.64. a0 = -0.1496; a2 = 0.1286; a4 = -0.0008; a6 = 0.0008 (6.66) By making this correction, an additional error is introduced in the equations (6.62). This additional error is equal to 0.1496 x 0.0027 = 0.0004. This additional error is small compared to the error h = -0.0338 (6.61) and will therefore be neglected. This concludes the approximation for the real part of r1 ej2s No replacement process is necessary because there were only five equations in the four-dimensional space. Next, the imaginary part of reej2's must be approximated.

88 From the tabulation (6.55) it can be seen that this imaginary part is very small for odd values of the variable 4s. According to Eq. 4.40: Im {G(s) e js} = n [ (s) + bneSne (s)] (6.67) The functions S2(s), S4(s), S6(s), and S8(s) will be chosen for the approximation process. The equations for the five sampling points can then be written using Table II: E4' -0.5000 b4 -0.1528 = 0 E5: -0.0606 b2 -0.2829 b4 +0.3472 b6 +0.1306 b8 -0.0490 = 0 El: 0.5000 b6 +0.2236 = O (6.68) E7: 0.0283 b2 +0.0772 b4 +0.2938 b6 -0 3395 b8 +0.0183 = 0 E8' -0.5000 b8 -0.2881 = 0 To determine the Chebyshev error for this system, one must first find coefficients X. such that: 4n4 + n5 + 6n+ + 7n7 + 7 8n8 = 0 (6.69) This leads to the following set of simultaneous linear equations: -0.0606 k5 +0.0283 X7 = 0 -0.5000 X4 -0.2829 k5 +0.0772 -7 = 0 (6.70) 0.3472 X5 +0.5000 +6 +0.2938 -7 = 0 0.1306 5 -0.3395 ~ -0.5000 5 8 = 0

89 The following values for Xj satisfy these equations: 4 = -0.1097; k5 = 0.4666; k6 = -0.9117; X7 = 1.0; k8 =-0.5572 (6 71) Using these values the Chebyshev error can be calculated: X4c4 + %5C5 + X6c6 + k7C7 + 8 (6 = -0.0102 (6.72) IX4l + Il51 + iX61 + 1x71 + Ix81 The error h. of equation E. is found from h. = h sgn Xj, and the following set of equations must then be solved to find the coefficients: E4. -0.5000 b4 -0.1528 = 0.0102 E5 -0.0606 b2 -0.2829 b4 +0.3472 b6 +0.1306 b8 -0.0490 = -0.0102 E6: 0.5000 b6 +0.2236 = 0.0102 E7: 0.0283 b2 +0.0772 b4 +0.2938 b6 -0.3395 b8 +0.0183 = -0.0102 E8: -0.5000 b8 -0.2881 = 0.0102 (6.73) The following values for the coefficients satisfy these equations: b2 = -2.8464; b4 = -0.3261; b6 = -0.4267; b8 = -0.5966 (6.74) This completes the synthesis of the matching section, The complete reflection-distribution function can now be written as follows: p(y) = - 0.1496 + 0.1286 cos 2icy - 0.0008 cos 4ny + 0.0008 cos 6gy - 2.8464 sin 2ny + 1.3181 sin 3-iy - 0.3261 sin 4ty + 0.4451 sin 5sy - 0.4267 sin 6iy + 0.4458 sin 7cy -0.5966 sin 8iy + 0.5363 sin 9ty (6.75)

90 This reflection-distribution function is plotted in Fig. 6.8. Using Eqs. 4.49 and 4.50, the characteristic-impedance function and the ratio of outer and inner conductor for a coaxial structure can be computed. The characteristic-impedance function Zo(y) is plotted in Fig. 6.9. Figure 6.10 shows a cross section of a matching section which has a reflection-distribution function equal to (6.75). As can be seen from Fig. 6.8, the reflection-distribution function has a rather large magnitude at a few points in the line. Also, the general behavior of p(y) is such that the characteristic impedance along the line varies between 8.5 and 85 ohms. The coefficient b2 = 2.8464 in the expansion (6.75) is largely responsible for this behavior. By choosing different approximating functions to form the imaginary part of G(s) ej2ts, a design can be obtained in which the characteristic impedance along the line does not vary over such a large range. This will be demonstrated below. The matching section, synthesized above, does not provide an exact match over the pass band, as is apparent from the Chebyshev errors found in Eqs. 6.61 and 6.72. These are the errors in the real and imaginary part of Cl(s) ej2As, respectively. To calculate the total reflection from the matching section, these two errors must be added vectorially. The total reflection is therefore equal to /0.03382 + 0.01022 = 0.0353, which corresponds to a VSWIR equal to 1.07, in the uniform line connecting the generator to the matching section. See Fig. 6.7. A general remark should be made regarding the total reflection error. In the example above, the total reflection error is equal at all five sampling points. This is a consequence of the fact that the number of sampling points exceeded the number of approximating functions

91 5 _ p(y) 4 2 — 1.25.5.75 1.0 -2 -4 Fig. 6.8 Reflection-distribution function for matching section No. 1. 100 Zo(y) - ( IN OHMS) 80 - 6040 20 0.25.5.75 1.0 Fig Characteristic function for matching section No Fig. 6.9 Characteristic-impedance function for matching section No. 1.

OUTSIDE CONDUCTOR CENTER CONDUCTOR a 60 cm Fig. 6.10 Cross section of matching section No. 1.

93 by exactly one. In general, when a large number of sampling points is taken, the maximum error in the real part will not necessarily occur at the same sampling points at which the error in the imaginary part is located. The only conclusion that can be drawn in such a case is that the total reflection error does not exceed the vectorial sum of the real and imaginary errors anywhere. To conclude this chapter, the imaginary part of 1r ej2is will be constructed again, using a different set of approximating functions. The approximating functions will be S4(s), S6(s), S8(s), and C5(s). Using Table I and Table II, the following equations can now be written for the sampling points: E4: -0.5000 b4 +0.2829 a5 -0.1528 = 0 E5: -0.2829 b4 +0.3472 b6 +0.1306 b8 +0.5000 a5 -0.0490 = 0 E: 0.5000 b6 +0.3472 a5 +0.2236 = 0 (6.76) E7: 0.0772 b4 +0.2938 b6 -0.3395 b8 +0.0183 = 0 E8 -0.5000 b8 -0.1306 a5 -0.2881 = 0 To find the Chebyshev error to this system, the set of X. must first be determined such that 4n4 5 + + 6n + 7 + n8 = (6.77) The coefficients Xj can be found by solving the following set of equations: -0.5000 k4 -0.2829 k5 +0.0772 X7 = o 0.3472 k5 +0.5000 k6 +0.2938 k7 = 0

0.1306 g5 -0.3395 7 -0.5000 ~8 = O 0.2829 k4 +0.5000 X5 +0.3472 -0.1306 = 0 ~~~~~~~5 ~~(6.78) The following values of Xj satisfy these equations: k4 = 1.2165; X5 = -2.8504; X6 = 3.4892; 7 = -2.5689; k8 = 1.0 (6.79) Using these values, the Chebyshev error for the system (6.76) can be found: 4c4 + 5c5 + 66 + X77 + 8 035(6.0) h = = 0.0358 (6. 80) i141 + 1i51 + ix61 + 1x71 + 1x81 The error hj of equation E. equals h. = h sgn X.. The errors can be substituted in the equations (6.76), resulting in: E4: -0.5000 b4 +0.2829a5 -0.1528 = 0.0358 E5: -0.2829 b4 +0.3472 b6 +0.1306 b8 +0.5000 a5 -0.0490 = -0.0358 E6: 0.5000 b6 +0.3472 a5 +0.2236 = 0.0358 E7' 0.0772 b4 +0.2938 b6 -0.3395 b8 +0.0183 = -0.0358 E8: -0.5000 b8 -0.1306 a5 -0.2881 = 0.0358 (6.81) The solution of these equations yields the following values for the coefficients: b4 = 0.6851; b6 = -1.6794; b8 = -1.1382; a5 = 1.8776 (6.82) This completes the synthesis of matching section No. 2. Using Eqs. 6.63 and 6.66, the reflection-distribution function can be written as:

95 p(Y) = - 0.1496 + 0.1286 cos 2xy - 0.0008 cos 4Ay + 1.8776 cos 5AY + 0.0008 cos 6ry + 1.3181 sin 31ry + 0.6851 sin 4vy + 0.4451 sin 5Ty - 1.6794 sin 6xy + 0.4458 sin 7ty - 1.1382 sin 8ty + 0.5363 sin 9gry (6.83) In Fig. 6.11 and 6.12, the real and imaginary parts of r1 eJ2ts and those of the approximation G(s) ej2Is are plotted. The approximation errors are those given by Eqs. 6.61 and 6.80. The total reflection at the input of matching section No. 2 is equal to the vectorial sum of 2 2 these errors: /0.0338 + 0.0358 = 0.0492. This corresponds to a VSWR of 1.10 in the uniform line connecting the generator in Fig. 6.7 to the load. This standing wave ratio is slightly higher than that obtained with matching section No. 1. Matching section No. 2, however, exhibits much smaller variations in the characteristic impedance along the line. The reflection-distribution function for this matching section is plotted in Fig. 6.13. Figure 6.14 shows the characteristic-impedance function. A cross section of the coaxial structure is shown in Fig. 6.15. It is of interest to determine the mechanical precision with which the nonuniform line shown in Fig. 6.15 has to be manufactured to produce the predicted result. An estimate of this tolerance can be made by considering the deviation allowed in the individual coefficients in the expansion (6.83) such that the resulting error in the input reflection of the line is an order of magnitude smaller than the input reflection itself. The input reflection coefficient of the line of Fig. 6.15 is equal to 0.05. One can then determine the deviation in the indiividual terms of the expansion (6.83) that would cause an error in the

96.3 F. 1 P.2, \ e {e2ap o 0 S 0 co~~o 1.0 1.25 1.75 2.0 -.1 -.3 Fig. 6.11 Plot of the function ReIm { ej2'S} and its approximation..3Im{Gej 27/s}.o 1.5 2.0 -.2

97 4 3 - 12 -3 90 Z0(y) C. (IN OHMS) -250 40 30.25.5.75 1.0 Fig. 6.13 Characteristic-impedance function for matching section No. 2.

OUTSIDE CONDUCTOR CENTER CONDUCTOR F _ _ I ~~~~~~~~~~~~~~ —_ ~ o 60 cm Fig. 6.15 Cross section of matching section No. 2.

99 input reflection coefficient equal to 0.005. An error in the coefficients of the expansion (6.83) will cause an error in the diameter of the center conductor of the line. When the inside diameter of the outside conductor (do) is equal to 1 inch, the deviations of the diameter of the center conductor, caused by the error in the coefficients of (6.83), are of the order of a few mills. For good results, therefore, the dimensions of the center conductor of the line have to be accurate to within a few thousandths of an inch.

CHAPTER VII CONCLUSIONS A general synthesis procedure has been developed for the synthesis of matching sections. The matching section provides a match between a generator, with complex internal impedance, and a complex load impedance, such that maximum power transfer is obtained over a given range of frequencies. Because the method is essentially a numerical one, the internal impedance of the generator and the Load impedance can be given either in equation form or in the form of measurements. A special case of a matching section is the impedance transformer which matches two real impedances of different values. The synthesis procedure can also be used, without essential modifications, for the synthesis of driving point impedances that must exhibit a certain behavior over a given band of frequencies. Nonuniform lines are synthesized which have the following properties: (a) The nonuniform line is of finite length (b) The line is lossless and has a homogeneous dielectric (c) The taper is continuous. The synthesis procedure is based on the approximate solution to the nonuniform-line equations that was developed by Orlov (Ref. 9) and Sharpe (Ref. 11). Using their approximate solution, the synthesis problem is first reduced to the problem of constructing a real function, identically zero outside a prescribed interval, whose Fourier transform approximates a given complex function. It is shown that this problem can be put into a convenient mathematical form when the reflection-distribution function is expanded in a trigonometric series. This results in 100

101 an approximation problem in which given functions have to be approximated by a linear combination of approximating functions. To solve this approximation problem the theory of discrete Chebyshev approximation is introduced and is shown to be a very powerful tool, excellently suited to the problem. Several examples have been given which demonstrate how impedance transformers and matching sections can be synthesized. It appears that the general synthesis procedure can be extended to the synthesis of nonuniform transmission lines, behaving as filters. A study of the cut-off characteristics of such filters would present an interesting area for further investigation. The magnitude of the reflection coefficient for filters, however, is no longer small compared to unity, and therefore methods must be developed by which the approximation errors can be evaluated. Another question that arises for possible further investigation is whether the approximate solution to the nonuniform-line equations can be extended to cover the case of lossy lines. If this were the case, the synthesis procedure developed in the present investigation might be extended to construct lossy nonuniform transmission lines also.

APPENDIX A METHOD FOR IMPROVED CONVERGENCE As was mentioned in the introduction to Chapter VI, it is possible to transform the set of functions Cn(s) and Sn(s) into a new set of functions having improved convergence. By improved convergence is meant that the side lobes of the new functions approach zero faster than those of the old functions Cn(s) and Sn(s), as the point s moves away from the main lobe. Use of the new functions will reduce the amount of computation, in case a large number of functions is used in the summation (4. 39). Danielson and Lanczos (Refs. 4 and 8) have published a transformation method'that can be adapted to the present problem. The four sets of functions, Cne(s), Cno(s), Sne(s), and Sno (s), have to be considered separately. It will be recalled that the symbol ne indicates only even values of n, and the symbol no stands for only odd values of n. First the summation Ne a C (s) (A.l) n=O ne ne is considered. It will be shown that the summation (A.1) equals the summation Ne Ne n a C (s) u U U(s) (A.2) when the coefficients a and u are related by the following transne ne formation: 102

103 u0 = a0 U2 = a2 + uO U - a + u (A.3) Une ne ne-2 UNe aNe + UNe-2 The transformation (A.3) can also be written in the following, equivalent form: ao = uo a0 = U2- U a u u (A 4) ne ne ne-2 (A.4) aNe = UNe - UNe-2 Substituting these values for a into (A.1), one finds: ne n a C (S) u0C0(s) + (u2 - u)C2(s) + *. + (Ue-Une2)Cne(s) + ~ ~ + (UNe - UNe.2)CNe(S) uO[CO(s)-C2(s)] + * * * + Une [Cne(s)-Cne+2(s)] + ~ ~ ~ (A.5)... + UNeCNe(S)

o04 By equating the corresponding terms in equations (A.2) and (A.5), it follows that the functions Une(s) are defined by the following relationship: Ue(s) - [C (s) - Cne+2(s)] 0 < ne < Ne (A.6) UNe(S) = CNe(s) The functions U ne(s) can be evaluated using equations (A.6) and (4.19). ne =..it 4s + n 4s + -n' 2 + n - 2 (A.-7) =... ( 4s+n)(4s+n+2 - (4s, n) (s-n-2) (n even) It is apparent from (A.7) that the functions U (s) converge faster than the functions C (s). The functions C (s), (s), s, and Sn (s) can be transformed in exactly the same manner as the functions C ne(s) above. The functions Cno (s) are transformed into the functions U no(s) such that No No n-1 anoCn(s) = l UnoUno(s) (A.8) where: U = a1 u3 = a3 + u1 (A.9) u a + U no no no-2 UNo = aNO + UNo-2

105 and the functions Uno (s) [Cno() - Cno2(S)] 1 < no < No (A.10) UNo(S) C (S) It follows from (A.10) and (4.20) n (S) -cos 2vs 2 2 no s ). 4Sns+n +2' (4s-n) (4s-n-2) (n odd) For the functions S ne(s), it is found that: Ne Ne Z b S (s) = 2 v V (s) (A.12) n=2 ne ne n2 ne where: v2 = b2 V4 = b4 + v2 (A. 13) v = b + v ne ne ne-2 V bN= b Ne-2 and the functions Vne(s) E [Sne(S) Sne+2(s) 2 < ne < Ne (A.14) VNe(S) - SNe(s)

106 Therefore, from (A.14) and (4.30), vne(s) i L 4s+n 4+n+2 - - + s-n-2 (A.15) sin 2s (4s+n)(4s+n+2) +4s-n)s-n-2 (n even) And finally, the functions S'(s) are transformed as follows: no No No 1 b S (s) = n v V (s) (A.16) no no no no where v = b1 V3 - b3 + 1 v = b + v(A.17) no bno o no-2 VNo bNo + VNo-2 and the functions Vno(S) equal V (s) [ () - S )s)] < no <No no no no1c(A.18) VNo(S) S o(S) And it follows from (A.18) and (4.31), that no (s) cos 2 [(4s+n)(4s+n+2) + (4s-n)(4s-n-2)] (A.19) (n odd)

LIST OF REFERENCES 1. E. Baur, "Beitrag zur Transformation mit inhomogenen Leitungen," Archiv der Elektrischen Ubertragung, XIII, No. 3, pp. 114-120, (March 1959). 2. E. F. Bolinder, "Fourier Transforms in the Theory of Inhomogeneous Transmission Lines" Proceedings IRE XXXVIII, No. 11, p. 1354, (November 1950). 3. E. F. Bolinder, "Fourier Transforms in the Theory of Inhomogeneous Transmission Lines," Acta Polytechnica, Electrical Engineering Series, III, No. 12, pp. 1-83, (19 51) 4. G. C. Danielson and C. Lanczos, "Some Improvements in Practical Fourier Analysis and. their Application to X-Ray Scattering from Liquids," Journal of the Franklin Institute, CCXXXIII, Nos. 4 and 5, pp. 365-380 and 435-452, (April and May 1942). 5. A. L. Feldshtein, "Nonuniform Lines as Filters," Radiotekhnika, VIII, No. 3, pp. 31-35, (1953). 6. A. L. Feldshtein, ".The Design of an Optimum Smooth Transition," Radiotekhnika. XIV, No. 3, pp. 40-46, (1959). 7. R. W. Klopfenstein, "A Transmission Line Taper of Improved Design," Proceedings IRE, XLIV, No. 1, pp. 31-35, (January 1956). 8. C. Lanczos, Applied Analysis, Prentice Hall, 1956. 9. S. I. Orlov, "Concerning the Theory of Inhomogeneous Transmission Lines," Zhurnal Tekhnicheskoi Fiziki, XXVI, No. 10, pp. 2361-2372, (October 1956). 10. H. Ruston, "Synthesis of R-L-C Networks by Discrete Tschebyscheff Approximations in the Time Domain," Ph.D. Thesis, Department of Electrical Engineering, The University of Michigan, April 1960. 11. C. B. Sharpe, "The Scattering Approach to the Synthesis of Nonuniform Lines," paper given at the North East Electronics Research and Engineering Meeting, Boston, Mass., November 14-17, 1960. An abstract is published in the Conference Digest. 12. E. Stiefel, "Uber diskrete und lineare Tschebyscheff-Approximationen," Numerische Mathematik, I, No. 1, pp. 1-28, (January 1959). 13. J. Willis and N. K. Sinha, "Non-uniform Transmission Lines as Impedance Transformers," Proceedings IEE, CIII (part B), pp. 166-172, (March 1956). 14. J. Willis and N. K. Sinha, "Impedance Transformers," Wireless Enineer, XXXIII, pp. 204-208, (September 1956). 107

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