SYNTHESIS OF MIXED LUMPED AND DISTRIBUTED ELEMENT NETWORKS Metin Şengül e-mail: [email protected] Kadir Has University, Engineering Faculty, Electronics Engineering Department, 34083, Cibali-Fatih, Istanbul, Turkey Key words: Synthesis, mixed lumped and distributed elements, two-variable networks, transfer scattering matrix ABSTRACT multivariable positive real functions, the problem of filter Synthesis algorithm for mixed lumped and distributed design with mixed lumped and distributed elements is element networks is presented. The algorithm is based attempted to be solved by many researcher especially on transfer scattering matrix factorization The mixed using the latter multivariable approach. In this context, structure is composed of ladder lumped-elements although there have been valuable contributions for the connected with commensurate transmission lines (Unit characterization of some restricted classes of mixed- elements, UEs). Reflection function expression in two- element structures, a complete theory for the variable is utilized in the algorithm. First, the type of approximation and synthesis problems of mixed lumped the element that will be extracted is determined. After and distributed networks is still not available. Thus, the obtaining the value of the element, it is extracted, and problem is quite challenging from a theoretical as well as the reflection function of the remaining network is the practical point of view. obtained by using transfer scattering matrix In the following section, the characterization of two- factorization method. Algorithm is given for low-pass variable networks is introduced. Subsequently, after structures (similar algorithms have been prepared for giving the synthesis algorithm, an example is presented, to high-pass, band-pass and band-stop structures). An illustrate the utilization of the proposed algorithm. example is included to illustrate the implementation of the proposed algorithm. II. CHARACTERIZATION OF TWO-VARIABLE NETWORKS I. INTRODUCTION An analytic treatment of the design problem with mixed lumped and distributed elements requires the Lossless S (p, ) R characterization of the mixed-element structures using 11 Two-Port transcendental or multivariable functions. The first approach deals with non rational single variable Figure 1. Lossless two-port with input reflectance transcendental functions and is based on the classical function S ( p,λ) . study of cascaded noncommensurate transmission lines by 11 Kinarivala [1]. First results on the synthesis of a transcendental driving-point impedance function as a Let {Skl ; k,l = 1,2} designate the scattering parameters of cascade of lumped, lossless, two-ports and commensurate a lossless two-port like the one depicted in Fig. 1. For a transmission lines were given by Riederer and Weinberg mixed lumped and distributed element, reciprocal, lossless [2]. The other approach to describe mixed lumped- two-port, the scattering parameters may be expressed in distributed two-ports is based on the transformation of Belevitch form as follows [5-8] Richards, λ = tanh( pτ ) which converts the S11 ( p, λ) S12 ( p, λ) transcendental functions of a distributed network into S( p, λ) = S 21 ( p, λ) S 22 ( p, λ) rational functions [3]. The attempts to generalize this (1a) approach to mixed lumped-distributed networks led to the 1 h( p, λ) µ f (− p,−λ) = multivariable synthesis procedures, where the Richards g( p, λ) f ( p, λ) − µh(− p,−λ) variable λ is used for distributed-elements and the f (− p,−λ) original frequency variable p for lumped-elements. In where µ = . f ( p,λ) this way, all the network functions could be written as rational functions of two or more complex variables. After In (1a), p = σ + jω is the usual complex frequency the pioneering work of Ozaki and Kasami [4] on the variable associated with lumped-elements, and λ = Σ + jΩ is the conventional Richards variable that f ( p,λ) of the entire mixed-element structure is in associated with equal length transmission lines or so product separable form as called commensurate transmission lines ( λ = tanh pτ , f ( p,λ) = f L ( p) f D (λ) , (1e) where τ is the commensurate delay of the distributed where f L ( p) and f D (λ) are the polynomials elements). constructed on the transmission zeros of the lumped- and th g( p,λ) is (n p + nλ ) degree scattering Hurwitz distributed-element subsections, respectively. polynomial with real coefficients such that In the lossless two-port network, if one only considers the T T T real frequency transmission zeros formed with lumped- g( p,λ) = P Λ g λ = λ Λ g P elements, then on the imaginary axis jω , f L ( p) will be where either an even or an odd polynomial in p . Furthermore, g g g 00 01 L 0nλ due to cascade connection of commensurate transmission T 2 np g10 g11 L M P = 1 p p p lines, f (λ) will have the following form Λ = , [ K ] D g n M M O M T 2 λ 2 nλ / 2 λ = []1 λ λ K λ . f D (λ) = (1− λ ) . (1f) g g np 0 L L npnλ A practical form of f ( p,λ) can be obtained by (1b) disregarding the finite imaginary axis zeros except those The partial degrees of two-variable g( p,λ) polynomial at DC as follows, are defined as the highest power of a variable, whose f ( p,λ) = p k (1− λ2 )nλ / 2 (1g) coefficient is nonzero, i.e. n p = deg p g( p,λ) , where k designate the total number of transmission zeros at DC. nλ = deg λ g( p,λ) . th Since the network is considered as a lossless two-port Similarly, h( p,λ) is also a (n p + nλ ) degree terminated in a resistance, then energy conversation polynomial with real coefficients such that requires that T T T T h()p,λ = P Λ h λ = λ Λ h P S( p,λ)S (− p,−λ) = I , (1h) where where I is the identity matrix. The open form of (1h) is h h h given as 00 01 L 0nλ T 2 np g( p, λ)g(− p,−λ) = h( p, λ)h(− p,−λ) + f ( p, λ) f (− p,−λ). h10 h11 L M P = 1 p p p Λ = , [ K ] (1i) h n M M O M T 2 λ λ = []1 λ λ K λ . In the following section, the synthesis algorithm, based on h h np 0 L L npnλ scattering transfer matrix factorization [9], for low-pass (1c) mixed-element two-port networks is given. The fundamental properties of this mixed-element structure f ( p,λ) is a real polynomial which includes all the can be found in [6] and [10]. transmission zeros of the two-port network. General form of the polynomial f ( p,λ) is given by III. MIXED-ELEMENT NETWORK SYNTHESIS i = 1,2,..., n ALGORITHM f ( p,λ) = f ( p) f (λ); (1d) In this section, synthesis algorithm for low-pass mixed- ∏ i j j = 1,2,..., m i, j element structures is explained step by step. Similar where n is the number of transmission zeros of the algorithms have been prepared for high-pass, band-pass lumped-element subsection ( n ≤ n ), the difference and band-stop structures, but because of space limitations, p they are not given here. ( n p − n ) is the number of transmission zeros at infinity of the lumped-element subsection, m is the number of 3.1. Low-Pass Ladders Connected with Unit Elements transmission zeros of the distributed-element subsection The coefficient matrices of the polynomials h( p,λ) and ( m ≤ nλ ), the difference ( nλ − m ) is the number of g( p,λ) , and polynomial f ( p,λ) describing the mixed- transmission zeros at infinity of the distributed-element element low-pass structure are as follows, subsection, fi ( p) and f j (λ) define the transmission zeros of lumped and distributed subsections, respectively. Transmission zeros can be located anywhere in p - and λ -planes. From (1d), it can be immediately deduced that the transmission zeros of each subsection have to arise in multiplication form. In other words, it can be assumed h00 h01 h02 L h0n g(n p ,0) + α1h(n p ,0) λ EV = . If the lumped element h h h h g(n −1,0) − α h(n −1,0) 10 11 12 L 1nλ p 1 p (L) (L) Λ h = h20 h21 L L 0 , is an inductor, the polynomials g ( p), h ( p) and 0 0 M M M (L) (L) EV (L) EV h ⋅ 0 0 f ( p) are g ( p) = p +1 , h ( p) = p , n p 0 L 2 2 (L) (RN ) (RN ) g g g g f ( p) = 1. The polynomials g ( p,λ), h ( p,λ) 00 01 02 L 0nλ g g g g and f (RN ) ( p,λ) of the remaining network can be 10 11 12 L 1nλ computed as Λ g = g 20 g 21 L L 0 , g (RN ) ( p,λ) = −h(L) (− p)h( p,λ) + g (L) (− p)g( p,λ) , M M 0 0 M g ⋅ 0 0 h(RN ) ( p,λ) = g (L) ( p)h( p,λ) − h(L) ( p)g( p,λ) , n p 0 L (RN ) 2 nλ / 2 f ( p, λ) = (1− λ2 ) nλ / 2 . f ( p,λ) = (1− λ ) . If the lumped element is a (C) (C) h(n ,0) capacitor, the polynomials g ( p), h ( p) and Step 1: Compute α = p = ±1 and 1 EV EV g(n p ,0) f (C) ( p) are g (C) ( p) = p +1 , h(C) ( p) = − p , h(n ,1) 2 2 p (C) (RN ) (RN ) α 2 = = ±1. If h(n p ,1) = 0 and g(n p ,1) = 0 , f ( p) = 1 .