1549-7747 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCSII.2018.2880262, IEEE Transactions on Circuits and Systems II: Express Briefs > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Abstract—In this work, the analysis of mixed-element structures formed with shunt capacitors separated by commensurate transmission lines is performed first time in the literature. Firstly a low-pass lumped-element ladder network is considered. Then the series inductors are replaced with commensurate transmission lines. As a result, a practically important mixed-element structure is obtained. Then the description of the structure by means of two frequency variables (one for shunt capacitors and one for transmission lines) is detailed: Explicit expressions for the coefficients of the descriptive two-variable polynomials in terms of the coefficients of the single variable boundary polynomials are derived for various numbers of elements, which are obtained first time in the literature. Finally a mixed-element broadband matching network is designed to illustrate the usage of the obtained expressions. If it is preferred not to have shunt capacitors, they can be replaced with open-ended stubs via Richard’s transformation. So the resultant circuit is extremely suitable for microstrip fabrication. Index Terms—Matching, mixed element networks, shunt capacitors, transmission lines I. INTRODUCTION HERE are lots of studies in the literature about ladder networks formed with inductors and capacitors [1-8]. While ladder networks with only lumped elements are studied in [1-3], low-pass ladder networks with lumped elements separated by transmission lines are given in [4]. High-pass, band-pass and band-stop versions are explained in [5]. The symmetric versions of these structures are studied in [6]. In [7], modeling based circuit design with these structures is given. Mixed element counterparts of the lumped element circuits are obtained in [8]. Usually it is not desired for the designed circuit to have inductors because they are heavy and bulky. Since they are available only for a limited range of values and are difficult to implement at microwave frequencies, they are approximated with distributed components. Richard’s transformation is used to convert lumped elements to transmission line sections, Manuscript received June 16, 2018; revised ???? ??, ????. This paper was recommended by X Y. This work was supported in part by Kadir Has University Scientific Research Project under Grant 2017-BAP-15. The authors are with the Faculty of Engineering and Natural Sciences, Kadir Has University, 34083 Cibali-Fatih-Istanbul, Turkey (e-mail: msengul@khas.edu.tr; gokhan.cakmak@stu.khas.edu.tr). while Kuroda’s identities can be used to separate circuit elements by using transmission line sections [9-12]. Now consider a low-pass lumped-element ladder network, a filter or a matching network. If we replace the series inductors between the shunt capacitors with transmission lines, we obtain a mixed-element structure. If the lengths of all the transmission lines are the same, these lines are called commensurate lines or unit elements (UE). It is very practical to fabricate this structure. If the transmission lines are quarter wavelength long, they are referred to as admittance inverters. These structures are useful especially for narrowband (<10%) bandpass and bandstop filters [9]. But in this work, it is not necessary to have quarter wavelength transmission lines. So it is possible to design broadband circuits. Also the transmission lines separating the shunt capacitors are not redundant elements; they are used as circuit elements effective for the desired response. Additionally if it is preferred not to have shunt capacitors, they can be replaced with open-ended stubs via Richard’s transformation. So the resultant circuit is extremely suitable for microstrip fabrication. II. TWO-VARIABLE DESCRIPTION OF LOSSLESS TWO-PORTS FORMED WITH CASCADED LUMPED AND DISTRIBUTED ELEMENTS Consider a lossless two-port formed with cascaded lumped and distributed-element sections. This structure can be described by using two-variable scattering parameters. The lumped and distributed-element sections are defined in terms of the classical frequency variable   j p   and the Richards variable     j  , respectively [8,13,14]. If the two-variable scattering matrix is ) , (  p S , then it can be written in terms of two-variable polynomials ) , (  p f , ) , (  p g and ) , (  p h as follows             ) , ( ) , ( ) , ( ) , ( ) , ( 1 ) , (         p h p f p f p h p g p S . (1) The polynomials ) , (  p f , ) , (  p g and ) , (  p h have the following properties:  ) , (  p f , ) , (  p g and ) , (  p h are polynomials with real coefficients.  ) , (  p g is a scattering Hurwitz polynomial.  ) , (  p f is formed via the transmission zeros of the Analysis of Mixed-Element Structures Formed with Shunt Capacitors Separated by Transmission Lines Metin Şengül and Gökhan Çakmak T