1
Background Theory and Concepts

Philippe Ferrari1, Marc Margalef-Rovira2, and Gustavo P. Rehder3

1TIMA, Université Grenoble Alpes, CNRS, Grenoble INP, Grenoble, France

2STMicroelectronics, RFC HDC, Crolles, France

3Polytechnic School, University of São Paulo, São Paulo, Brazil

The objective of this introductory chapter is to draw up a brief history of slow-wave structures in Section 1.1, then to define the concept of slow-wave propagation from a theoretical point of view in Section 1.2, next to briefly present the three slow-wave transmission lines (in Section 1.3) presented in detail within the following chapters, namely slow-wave coplanar waveguides (S-CPWs, Chapter 2), slow-wave coplanar striplines (S-CPS, Chapter 2), slow-wave microstrip (S-MS, Chapter 3), slow-wave substrate integrated waveguides (SW-SIW, Chapter 4), and finally, in Section 1.4, to highlight some advantages of modern slow-wave transmission lines, which is a big part of the motivation for this book.

1.1 Historical Background

During the second half of the 20th century, technological advancements had greatly impacted modern society, with telecommunication networks being one of the most notable. With numerous scientific and technological breakthroughs, these networks have become more complex and efficient. This has led the consumer electronics market to become economically significant, with the development of new services and activities driven by the increasing demand for high-definition multimedia applications, secure data transmission, wearables, etc. To provide the necessary bandwidths and subsequent data rates for these applications, the next generation of wireless communications is oriented toward higher frequencies, especially the millimeter wave (mm-wave) bands. However, this presents new challenges, such as the need for wireless transceiver circuits that can operate at high frequencies with reasonable efficiencies, relying on low-cost technologies and compact solutions. Slow-wave structures can be a solution for the design of compact circuits.

The concept of slow-wave structures emerged in the early 1940s, for their capability to establish efficient interaction with electron beams. More precisely, it started in the context of radar applications where these interactions were used to amplify RF waves. The amplification was based on transferring kinetic energy from electrons to a propagating wave. Based on this principle, the first slow-wave structure was called "klystron," a high-frequency vacuum tube invented in 1937 by W. Hansen and the Varian brothers (Varian & Varian, 1939), as illustrated in Fig. 1.1(a). In these amplifiers, the slow-wave propagation was achieved by cascading resonant cavities, which resulted in narrowband operation (Wu, 1999). In these structures, the slow-wave propagation was necessary because the interaction requires close velocities between the wave and the electrons, which move in vacuum at a lower velocity than the light. This interaction was further enhanced by R. Kompfner, who realized, in 1943, a broadband amplifier based on a nonresonant helix structure called "traveling-wave tube" (TWT; Kompfner, 1947), illustrated in Fig. 1.1(b). Improved versions of these devices are still in use for radar, satellite communications, television broadcasting, and particle accelerators. Oscillators have also been developed based on the same principles.

Figure 1.1 (a) First commercial klystron.

Source: Henney 1940/with permission of WorldRadioHistory;

(b) Traveling-wave tube principle of operation.

Source: Adapted from Kompfner (1947).

During the 1960s, the development of integrated microwave circuits provided a good opportunity to develop layered structures with potential slow-wave propagation. This concept was demonstrated for the first time in 1969 for a metal-insulator-semiconductor microstrip structure (Hasegawa et al., 1971) on silicon. It was followed by several topologies, including the Schottky contact transmission line (Jager, 1976). Thanks to an external bias, this last structure was used to create a variable slow-wave effect (Jaffe, 1972). As explained in (Wu, 1999), planar periodic structures gained attention in the early 1970s for the development of wide-band coupled microstrip lines (Podell, 1970). Since then, the research on planar periodic and layered structures has continued until today. In the meantime, slow-wave planar structures have also been used for miniaturization purpose. In microwave passive circuit design, specific functions such as filters, antennas, and couplers can be realized by the combination of physical phenomena such as interference, resonance, and couplings. These phenomena are very often dependent on wavelengths, so that specific properties can be obtained for given dimensions. It also means that, in general, these passive circuits occupy much larger areas than the active ones, which are made of increasingly smaller transistors. For example, a floating straight transmission line has intrinsic resonance frequencies that are directly related to the ratio of the propagation velocity and its physical length. Therefore, for a given frequency, a miniaturized structure can be obtained if the velocity is accordingly reduced. Obviously, the challenge lies in the effort to make such slow-wave structures as efficient in terms of dissipation as the original ones. In printed-circuit-board technology, a high number of topologies have been developed in the recent years, some of them are illustrated in Fig. 1.2. Figure 1.2(a) shows spoof surface plasmon-based transmission lines, for compact designs, reduced attenuation and limited cross-talk (Kianinejad et al., 2015). Compact couplers, such as the "rat-race," are illustrated in Fig. 1.2(b). The compactness was achieved by using a high slow-wave factor (SWF) microstrip structure (Chang & Chang, 2012). In Fig. 1.2(c), a filter based on six slow-wave resonators is also shown (Shi et al., 2010). One could also mention the use of defected ground (Kim & Lee, 2006) and electromagnetic band-gap (Zhurbenko et al., 2006), which often exhibit slow-wave propagation.

Figure 1.2 (a) Spoof surface plasmon-based slow-wave transmission lines.

Source: Kianinejad et al. (2015)/with permission of IEEE;

(b) Rat-race coupler.

Source: Wei-Shin Chang et al. 2012/with permission of IEEE;

(c) Six-resonator low-pass filter based on slow-wave resonators.

Source: Shi et al. (2010)/with permission of IEEE.

Concerning integrated technologies, the design of compact and low-loss passive circuits is a real challenge. It is especially true for the newly addressed mm-wave bands, in which parasitic couplings are more and more limiting and where the high conduction losses result in poor quality factors. In this context, slow-wave structures do not only provide miniaturized circuits but may also lead to higher quality factors (Chee et al., 2006; Cheung & Long, 2006; Franc et al., 2013). This is the case of the S-CPW, whose geometry prevents conduction losses in the semiconductor by shielding the electric field (see Fig. 1.3(a)). Meander lines were also used to realize compact couplers in silicon-based integrated passive device (IPD) technologies (Tseng & Chen, 2016), as shown in Fig. 1.3(b). A band-pass filter using a slow-wave microstrip topology was presented in (Evans et al., 2012), it is illustrated in Fig. 1.3(c).

1.2 The Slow-Wave Concept

In this section, a general theoretical approach explaining the concept of slow-wave propagation is presented. It is based on the magnetic and electric energies that are stored in a waveguide (Bertrand et al., 2020).

A general uniform waveguide topology is illustrated in Fig. 1.4.

The cross section could contain either different metallic conductors, magnetic, or dielectric materials. A wave propagating inside such a waveguide is characterized by its phase constant ß and angular frequency ?. By definition, its phase velocity vp is the velocity at which the phase of the wave travels in space, and is defined as (1.1).

Figure 1.3 (a) Slow-wave coplanar waveguide topology.

Source: Franc et al. (2013)/with permission of IEEE;

(b) Slow-wave coupler in silicon-based IPD technology.

Source: Tseng et al. 2016/with permission of IEEE;

(c) Miniaturized slow-wave microstrip filter in 65-nm CMOS technology.

Source: Evans et al. (2012)/with permission of The Institution of Engineering and Technology.

It can be seen as the velocity at which an observer should travel along the waveguide in order to keep in state with this wave. A second velocity is called group velocity, vg, and it is the velocity at which the overall shape of the waves' amplitudes - or modulation - travels through space. This velocity is also often interpreted as the velocity at which the energy or information propagates; it is given by (1.2).