IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 29, NO. 7, JULY 2019 465 Dispersive Delay Structure Based on Interdigital Capacitors With Noncommensurate Fingers Hossein Nazemi-Rafi and Masoud Movahhedi , Member, IEEE Abstract— Interdigital capacitors are key components in many RF and microwave designs such as oscillators, filters, and dispersive delay structures (DDSs). A conventional interdigital capacitor has equal length fingers and has a straight shaped figure. In this letter, we introduce an interdigital capacitor with noncommensurate fingers with a curved shape and propose a modified approach for analyzing this kind of capacitors based on N noncommensurate coupled-lines network. Moreover, this capacitor is used as unit cells to create a novel DDS for analog signal processing (ASP) applications. The proposed DDS creates extremely high group delay difference of 30 ns between 1.25 and 1.75 GHz and has a compact size of 0.6 × 0.6λ 2 and low insertion loss. Index Terms— Dispersive delay structure (DDS), group delay (GD), interdigital capacitor, metamaterial. I. I NTRODUCTION I NTERDIGITAL capacitors are one of the popular elements for designing and developing RF and microwave networks. These components have been analyzed by She and Chow [1] and Dib et al. [2]. A conventional interdigital capacitor has equal length fingers and can be modeled as a network with 4 or 2 ports. Dib et al. [2] introduced a new model for ana- lyzing interdigital capacitors, which was based on N-coupled lines network combined with two connection parts [2]. In this letter, their model is employed and modified to analyze the interdigital capacitor with noncommensurate fingers as a unit cell. Then, these unit cells are cascaded to create an inter- digital transmission line (ITL) [3] and used as a dispersive delay structure (DDS). Due to periodic capacitive loading, the interdigital DDS based on these unit cells performs similar to slow-wave structures [4]. DDS is a basic component in some analog signal processing (ASP) systems. The phase of any DDS, i.e., φ(ω), is a nonlinear function of frequency, so its group delay (GD), i.e., τ(ω) =-d φ/d ω is a frequency- dependent parameter [5]. II. I NTERDIGITAL CAPACITOR Fig. 1 shows schematics for interdigital capacitors with commensurate and noncommensurate fingers. An interdigital capacitor with commensurate fingers can be separated into three parts; one N -coupled lines network and two connection structures [2]. The similar method can be used for ana- lyzing interdigital capacitor with noncommensurate fingers. Manuscript received April 14, 2019; revised April 29, 2019; accepted May 5, 2019. Date of publication May 30, 2019; date of current version July 3, 2019. (Corresponding author: Masoud Movahhedi.) The authors are with the Department of Electrical Engineering, Yazd University, Yazd 89195-741, Iran (e-mail: nazemi.hossein@stu.yazd.ac.ir; movahhedi@yazd.ac.ir). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LMWC.2019.2915920 Fig. 1. (a) Commensurate and (b) noncommensurate interdigital capacitor. Fig. 2. Noncommensurate coupled-lines network. Fig. 3. Separating the noncommensurate coupled-lines network to 2N -1 commensurate coupled-lines subnetwork. The first step is analyzing N noncommensurate coupled-lines network [6]. Fig. 2 shows the N -coupled lines network, with each line is shorter than the line above. Assume that the length reduction is uniform, therefore, the length of the j th line of the network is ` j = ` 1 - 2( j - 1)1x , j = 1,..., N (1) where 1x is the value of length reduction from one side of the network. This network can be separated into (2 N -1) subnetworks with commensurate coupled lines [6] as shown in Fig. 3. To obtain the ABCD matrix of the network, the ABCD matrix of each subnetwork should be multiplied to the next one. However, the dimension of each ABCD matrix is different and varies from (2 × 2) to (2 N × 2 N ). Imaginary lines with ideal transmission properties can be added to resolve this problem [6]. Fig. 4 shows the completed subnetwork with imaginary ideal lines. For the j th subnetwork ( j = 1,..., 2 N - 1) of Fig. 3, the ABCD matrix can be found as [6] ABCD j =  cos(β1x )I j j ωL sin(β1x )/β j ωC sin(β1x )/β cos(β1x )I j  (2) 1531-1309 © 2019 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.