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 . The polynomials g ( p,λ), h ( p,λ) g(n p ,1) and f (RN ) ( p,λ) of the remaining network can be h(n p −1,1) α = = ±1. computed as 2 g(n −1,1) p g (RN ) ( p,λ) = −h(C) (− p)h( p,λ) + g (C) (− p)g( p,λ) , Step 2: h(RN ) ( p,λ) = g (C) ( p)h( p,λ) − h(C) ( p)g( p,λ) , α1 α 2 First component Next component (RN ) 2 n / 2 +1 -1 UE Series inductor (L) f ( p,λ) = (1− λ ) λ . Parallel capacitor Step 4: Set new h( p,λ) , g( p,λ) and f ( p,λ) two- -1 +1 UE (C) variable polynomials as h( p,λ) = h(RN ) ( p,λ) , +1 +1 Series inductor (L) UE (RN ) (RN ) Parallel capacitor g( p,λ) = g ( p,λ) , f ( p,λ) = f ( p,λ) , and go to -1 -1 UE (C) Step 1. Step 3: a) If the component that will be extracted is a UE, IV. EXAMPLE 1+ S λ then characteristic impedance is Z = 11 , where The coefficient matrices of the polynomials h( p,λ) and 1− S λ 11 g( p,λ) , and polynomial f ( p,λ) describing the mixed- nλ h(0,i) element low-pass structure are given as, ∑ 0 3.15 −1.05 1 3.85 1.45 S λ = i=0 . The polynomials g (UE) (λ), h (UE) (λ) 11 n λ 3.5 − 3.8 23.3 6.5 14 23.3 g(0,i) Λ h = , Λ g = ∑ 3 65.4 0 15 65.4 0 i=0 2 36 0 0 36 0 0 (UE) (UE) Z +1 and f (λ) are g (λ) = λ +1, 2 2Z , f ( p, λ) = (1− λ ). 2 h(n p ,0) h(3,0) 36 (UE) Z −1 (UE) 2 1/ 2 Step 1: α1 = = = = +1. h(3,1) = 0 h (λ) = λ , f (λ) = (1− λ ) . The g(n p ,0) g(3,0) 36 2Z and g(3,1) = 0 ⇒ polynomials g (RN ) ( p, λ) , h (RN ) ( p, λ) and f (RN ) ( p,λ) h(n p −1,1) h(2,1) 65.4 of the remaining network can be computed as α = = = = +1. 2 g(n −1,1) g(2,1) 65.4 g (RN ) ( p,λ) = −h(UE) (−λ)h( p,λ) + g (UE) (−λ)g( p,λ) , p Step 2: α1 = +1 and α 2 = +1, so the first component that h(RN ) ( p,λ) = g (UE) (λ)h( p,λ) − h(UE) (λ)g( p,λ) , will be extracted is an inductor, and the element value is f (RN ) ( p,λ) = (1− λ2 )(nλ −1) / 2 . b) If the component that will be extracted is a lumped element (L or C), then the element value is g(n p ,0) +α1h(n p ,0) g(3,0) + h(3,0) But a complete theory for the approximation and synthesis EV = = problems of mixed-element networks is still not available. g(n −1,0) −α h(n −1,0) g(2,0) − h(2,0) p 1 p , So in this paper, synthesis of practically important class of 36 + 36 mixed-element networks, namely ladder lumped-elements = = 6 15 − 3 connected with UEs, is examined. The synthesis of mixed-element networks may be realized by using single L = 6 . The polynomials g (L) ( p), h(L) ( p) and f (L) ( p) 1 variable boundary polynomials, namely the polynomials EV of the inductor are g (L) ( p) = p +1 = 3p +1, h( p,0), g( p,0), f ( p,0) for lumped-element section, and 2 h(0,λ), g(0,λ), f (0,λ) for distributed-element section. EV h(L) ( p) = p = 3p , f (L) ( p) = 1. The polynomial In this case, synthesis is carried out for lumped and 2 distributed sections separately. Then, the components are (RN ) f ( p,λ) and coefficient matrices Λ h and Λ g of the mixed, to construct the mixed-element network. But in the proposed algorithms, the synthesis of mixed-element remaining network are f (RN ) ( p,λ) = (1− λ2 ) , network is carried out by using the two-variable reflection 0 3.15 −1.05 1 3.85 1.45 function of the mixed-element network, and components are extracted according to the connection order in the Λ h = 0.5 − 5.9 15.8 , Λ g = 3.5 11.9 15.8 . mixed-structure. Synthesis algorithm, based on transfer − 6 12 0 6 12 0 scattering matrix factorization, for low-pass networks is If the same algorithm is used, the extracted element values given. The implementation of the proposed algorithm is and the remaining network coefficient matrices and utilized by the given example. As a result, a simple to use polynomial f (RN ) ( p,λ) are as follows, mixed-element network synthesis algorithm is presented, Z of UE =2 and which is necessary for the applications using this 1 1 important class of mixed-element networks, e.g. design of 0 2.4 1 2.6 filters, broadband matching networks and amplifiers. Λ h = 0.5 7.9 , Λ g = 3.5 7.9 , − 6 0 6 0 REFERENCES 1. B.K. Kinariwala, Theory of cascaded structures: , f (RN ) ( p, λ) = (1− λ2 )1/ 2 , lossless transmission lines, Bell Syst. Tech. J., 45, pp.631-649, 1966. 0 2.4 1 2.6 C1 = 3 and Λ h = , Λ g = , 2. L.A. Riederer, L. Weinberg, Synthesis of lumped- 2 − 0.4 2 0.4 distributed networks: lossless cascades, IEEE Trans. f (RN ) ( p,λ) = (1− λ2 )1/ 2 , CAS, 27, pp.943-956, 1980. 3. P.I. Richards, Resistor transmission line circuits, Z of UE =5 and 2 2 Proc IRE, 36, pp.217-220, 1948. 0 1 (RN ) 4. H. Ozaki, T. Kasami, Positive real functions of Λ h = , Λ g = , f ( p,λ) = 1, 2 2 several variables and their applications to variable (RN ) networks, IEEE Trans. Circuit Theory, 7, pp.251- L2 = 4 and Λ h = 0 , Λ g = 1 , f ( p,λ) = 1, 260, 1960. and the termination resistor R = 1 . The obtained network 5. B.S. Yarman, A. Aksen, An integrated design tool to is given in Fig. 2. construct lossless matching networks with mixed lumped and distributed elements, IEEE Trans. CAS, L1 L2 39, pp.713-723, 1992. 6. A. Aksen, Design of lossless two-port with mixed, lumped and distributed elements for broadband Z1 Z2 R C matching, PhD Thesis, Ruhr University, 1994. 1 7. A. Aksen, B.S. Yarman, A real frequency approach
Figure 2. Synthesized low-pass mixed-element network, to describe lossless two-ports formed with mixed normalized element values: L = 6, L = 4, C = 3, lumped and distributed elements, Int. J. Elec. Comm. 1 2 1 (AEÜ), 55, (6), pp.389-396, 2001. Z1 = 2, Z 2 = 5, R = 1 . 8. A. Sertbaş, B.S. Yarman, A computer aided design technique for lossless matching networks with mixed V. CONCLUSION lumped and distributed elements, Int. J. Elec. Comm. The unavoidable connections between lumped-elements (AEÜ), 58, pp.424-428, 2004. destroy the performance of the lumped-element networks 9. A. Fetweiss, Cascade synthesis of lossless two-ports at high frequencies. But these connection lines can be by transfer matrix factorization, in R. Boite: used as circuit components. In this case, the circuits must Network theory, pp.43-103, Gordon&Breach, 1972. be composed of mixed lumped and distributed elements. 10. A. Sertbaş, Genelleştirilmiş iki-kapılı, iki-değişkenli kayıpsız merdiven devrelerin tanımlanması, PhD Thesis, İstanbul University, 1997.
Metin Şengül received B.Sc. and M.Sc. degrees in Electronics Engineering from İstanbul University, Turkey in 1996 and 1999, respectively. He completed his Ph.D. in 2006 at Işık University, İstanbul, Turkey. He worked as a technician at İstanbul University from 1990 to 1997. He was a circuit design engineer at R&D Labs of the Prime Ministry Office of Turkey between 1997 and 2000. Since 2000, he is a lecturer at Kadir Has University, İstanbul, Turkey. Currently he is working on microwave matching networks/amplifiers, data modeling and circuit design via modeling. Dr. Şengül was a visiting researcher at Institute for Information Technology, Technische Universität Ilmenau, Ilmenau, Germany in 2006 for six months.