**2. Uniplanar lines and multimodal models for transitions and discontinuities**

#### **2.1. The slotline and the coplanar waveguide**

**1. Introduction**

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technology.

MEMS-reconfigurable RF communication circuits [18].

which depends on the membrane dimensions [24].

Radio frequency microelectromechanical system (RF-MEMS) switches are aimed to perform the control function in tunable and reconfigurable RF/microwave and millimeter-wave (mmwave) systems. Electrostatic actuation is often preferred to other actuation mechanisms like electrothermal [1, 2] and phase-change/phase-transition materials [3], due to its negligible current consumption, no requirement for external heating sources and integration capability with well-established technologies such as high-resistivity silicon [4–7], fused-quartz and glass substrates [8–11], or CMOS [12–15] and SiGe BiCMOS [16, 17] processes. The latter can provide totally integrated, efficient systems containing sensors, control electronics, and

The mechanical and electrical design of the RF-MEMS switches has been comprehensively studied in the literature [19], and it highly depends on the circuit or transmission media in which it is to be integrated and the technology platform [20]. A number of solutions can be found, including integration in microstrip transmission lines [21], coplanar waveguides (CPWs) [4], coplanar striplines (CPSs) and slotlines [11], planar structures embedded in rectangular waveguides [22], and micromachined waveguides for sub-mm-wave frequencies [23]. Depending on the specific designs and dimensions, they can operate in the microwave and the mm-wave bands, at frequencies as high as 240 GHz as reported in [24] using BEOL in BiCMOS

Series and parallel RF-MEMS switch topologies can be implemented, with either ohmiccontact [22] or capacitive-contact [4, 25]. While ohmic switches can operate in a very wide frequency band from DC to mm-waves featuring excellent OFF-state isolation and very low ON-state insertion loss, capacitive switches are frequency selective (being the center frequency defined by a series LC-resonant circuit) but their operation can be extended well beyond mmwave frequencies by properly choosing the ON-state capacitance and the series inductance

Mechanical topologies for RF-MEMS switches include bridge-type clamped-clamped or beamsuspension membranes and cantilever-type switches. Important switch parameters, such as the actuation voltage or the fabrication residual stress, are dependent on the particular selected topology [26–28]. Using three-dimensional (3D) mechanical simulation, the material physical properties are taken into account to a priori assess the behavior of the switch geometry (including the suspension type) in terms of initial membrane deformation, pull-in voltage, spring constant, mechanical resonant frequency, and transition times from OFF to ON states (and vice versa) [29]. Mechanical transients may produce bouncing phenomena [30–34] which degrade the RF behavior of the switch and can be studied more efficiently with energy models [35].

RF-MEMS switches featuring the above mechanical topologies are compatible with and can be conveniently integrated in uniplanar structures (CPW, CPS, and slotline) to perform a control function. In case of multimodal transmission lines like CPW, they can be used to selectively control the two CPW fundamental propagation modes (even and odd) [36]. To accurately analyze the interaction between modes in complex uniplanar structures (transitions, discontinuities), multimodal circuit models are derived from the application of the general multimodal theory [37–40]. Moreover, suitable equivalent circuits for both (ON/OFF) states The slotline and the CPW are uniplanar transmission lines. The slotline consists of two conductor strips on a dielectric substrate (**Figure 1(a)**). The CPW consists of three conductor strips on a dielectric substrate (**Figure 1(b)**). The slotline is a monomodal transmission line: it propagates only one fundamental quasi-transversal electromagnetic (TEM) mode, whose voltages and currents (both for the total voltage and current *Vs* (*z*) and *I s* (*z*), and for the forward and backward propagating waves *Vs* + (*z*), *I s* + (*z*), and *Vs* − (*z*), *I s* − (*z*), respectively) are defined as in **Figure 2(a)** and can be circuitally modeled as an ideal transmission line (**Figure 2(b)**), with

and backward propagation waves  $V\_{\varsigma}^{\*}(\mathbf{z})$ ,  $I\_{\varsigma}^{\*}(\mathbf{z})$ , and  $V\_{\varsigma}^{\*}(\mathbf{z})$ ,  $I\_{\varsigma}^{\*}(\mathbf{z})$ , respectively) are

$$V\_{\varsigma}(\mathbf{z}) = V\_{\varsigma}^{\*}(\mathbf{z}) + V\_{\varsigma}^{-}(\mathbf{z}) \qquad V\_{\varsigma}^{\*}(\mathbf{z}) = V\_{\varsigma}^{\*}e^{-\beta\bar{\varsigma}z} \qquad V\_{\varsigma}^{-}(\mathbf{z}) = V\_{\varsigma}^{-}e^{+\beta\bar{\varsigma}z}$$

$$I\_{\varsigma}(\mathbf{z}) = I\_{\varsigma}^{\*}(\mathbf{z}) + I\_{\varsigma}^{-}(\mathbf{z}) \qquad I\_{\varsigma}^{\*}(\mathbf{z}) = V\_{\varsigma}^{\*}(\mathbf{z})/Z\_{0\varsigma} \qquad I\_{\varsigma}^{-}(\mathbf{z}) = -V\_{\varsigma}^{-}(\mathbf{z})/Z\_{0\varsigma}$$

where *Z*0*<sup>s</sup>* is the characteristic impedance of the slotline mode and *β<sup>s</sup>* its phase constant. The CPW is a multimodal transmission line: it can propagate two fundamental quasi-TEM modes simultaneously (the even and odd modes) whose voltages and currents are defined as in **Figure 3(a)**. The odd mode is often seen as spurious, and its propagation cut by means of air bridges (described subsequently). However, it can be used to design new kinds of compact uniplanar circuits. In a CPW section, the even and odd modes do not interact between them and therefore can be circuitally modeled as two independent ideal transmission lines, as shown in **Figure 3(b)**, with equations analogous to those of the slotline for either of them.

example, some simple multimodal models are presented subsequently; more complex ones

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The layout of this transition is shown in **Figure 4(a)** (the depicted voltages and currents are the total ones for each mode, computed at the transition plane). Its behavior is easy to understand intuitively. At the transition, the odd mode transforms into the slotline mode and vice versa due to their similarity of voltage and current orientations (caused by the similarity of their electromagnetic fields). The even mode, however, is left in open circuit when the slotline begins since its current in the CPW central strip can flow no more. Therefore, the multimodal circuit model

**Figure 3.** (a) Definitions of voltages and currents in a CPW. (b) Circuit model for a CPW section.

**Figure 4.** (a) Symmetric CPW-to-slotline transition. (b) Multimodal circuit model.

are described in [6, 9–11, 38–40, 58].

*2.2.1. Symmetric CPW-to-slotline transition*

#### **2.2. Multimodal models for CPW circuits**

The even and odd modes behave differently when they encounter any transition or asymmetry, and there they may also interact between them. A multimodal model is a circuit model that makes the behavior of the different modes at a transition or asymmetry explicit. As an

**Figure 1.** (a) Slotline. (b) CPW.

**Figure 2.** (a) Definitions of voltages and currents in a slotline. (b) Circuit model for a slotline section.

example, some simple multimodal models are presented subsequently; more complex ones are described in [6, 9–11, 38–40, 58].

#### *2.2.1. Symmetric CPW-to-slotline transition*

The CPW is a multimodal transmission line: it can propagate two fundamental quasi-TEM modes simultaneously (the even and odd modes) whose voltages and currents are defined as in **Figure 3(a)**. The odd mode is often seen as spurious, and its propagation cut by means of air bridges (described subsequently). However, it can be used to design new kinds of compact uniplanar circuits. In a CPW section, the even and odd modes do not interact between them and therefore can be circuitally modeled as two independent ideal transmission lines, as shown in **Figure 3(b)**, with equations analogous to those of the slotline for either of them.

The even and odd modes behave differently when they encounter any transition or asymmetry, and there they may also interact between them. A multimodal model is a circuit model that makes the behavior of the different modes at a transition or asymmetry explicit. As an

**Figure 2.** (a) Definitions of voltages and currents in a slotline. (b) Circuit model for a slotline section.

**2.2. Multimodal models for CPW circuits**

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**Figure 1.** (a) Slotline. (b) CPW.

The layout of this transition is shown in **Figure 4(a)** (the depicted voltages and currents are the total ones for each mode, computed at the transition plane). Its behavior is easy to understand intuitively. At the transition, the odd mode transforms into the slotline mode and vice versa due to their similarity of voltage and current orientations (caused by the similarity of their electromagnetic fields). The even mode, however, is left in open circuit when the slotline begins since its current in the CPW central strip can flow no more. Therefore, the multimodal circuit model

**Figure 3.** (a) Definitions of voltages and currents in a CPW. (b) Circuit model for a CPW section.

**Figure 4.** (a) Symmetric CPW-to-slotline transition. (b) Multimodal circuit model.

for the symmetric CPW-to-slotline transition is that of **Figure 4(b)**. As can be seen, a multimodal circuit model confines the contributions of each mode present in a transition into a different port.

#### *2.2.2. Impedances connecting the two outer CPW strips and air bridges*

Suppose an impedance connects the two outer CPW strips as shown in **Figure 5(a)**. This circuit can model an air bridge (a conducting wire connecting the two outer CPW strips, with an impedance *Z* = 0 ideally), but also more complicated situations. The even mode does not interact with the impedance since the two outer CPW conductors have the same even-mode electric potential. Therefore, the impedance behaves as a shunt impedance for the odd mode, and it is transparent to the even mode. Thus, its circuit model is that of **Figure 5(b)**. As can be seen, an air bridge (*Z* = 0) blocks the propagation of the odd mode by short-circuiting it. By controlling the value of *Z*, for instance, by means of MEMS switches, the amount of odd mode that propagates from the left side of the CPW to the right one can be controlled without affecting the propagation of the even mode.

**3. Fabrication technology for electrostatically actuated RF-MEMS** 

**Figure 6.** (a) Asymmetric shunt impedances connecting the strips of a CPW. (b) Multimodal circuit model.

A flexible technology platform has been developed and optimized at the FBK Institute (Trento, Italy) for the fabrication of RF-MEMS. Basic components (like low-loss CPW, microstrip line and slotline, ohmic [41, 42] and capacitive [43, 44] switches, variable capacitors and inductors) can be integrated in complex reconfigurable RF circuits. Many kinds of devices were produced, mainly for space and communication application, like switching matrices [45, 46], tunable and switchable phase shifters [47], reconfigurable antennas, impedance matching networks [48], VCOs [49, 50], and tunable filters. Depending on the used substrate, highresistivity silicon (<40 GHz), or fused quartz (>40 GHz), the working frequency range spans

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The base process requires eight lithography masks but, depending on the requirements, it can easily be expanded to deposit and pattern metal on the wafer backside to realize microstrip lines or antennas and to obtain devices suspended over thin membranes by locally removing the substrate. A wafer-to-wafer or a cap-to-die-bonding module is also available to encapsu-

RF signal lines and ground area are made of thick electroplated gold to reduce insertion losses while actuation electrodes and DC-bias signal lines are made of a high-resistivity polysilicon to minimize coupling with adjacent RF lines. The movable and suspended structures of the electrostatically actuated switches, which can be either cantilevers or clamped-clamped beams, are made by gold deposited over a sacrificial photoresist layer having the thickness of the required air gap, while switch underpass lines and other conductors are made of a thin Al film. On ohmic-contact switches, the gold-to-gold contact area is defined by underneath polysilicon protruding dimples to ensure a repeatable contact force and a uniform and reproducible low contact resistance. On capacitive-contact switches, the contact capacitor is made by depositing a thin silicon oxide dielectric and an upper floating metal (FLOMET) electrode

**switches**

**3.1. RF-MEMS technology platform**

from sub-GHz up to more than 100 GHz.

late the delicate MEMS moving parts [51].

#### *2.2.3. Asymmetric shunt impedances in a CPW*

In the two previous examples, the even and odd modes behaved in a different way at the analyzed transitions but did not interact between them due to the symmetry of the transitions. When the transitions are asymmetric, as it is the case for the asymmetric shunt impedances connecting the strips of the CPW shown in **Figure 6(a)**, the modes interact between them. The behavior of this transition is not obvious, but it can be rigorously modeled by the circuit shown in **Figure 6(b)** [37]. As can be seen, in this case, there is an energy balance between even and odd modes (there is a circuit connection between the even- and odd-mode ports), provided that the impedances *ZA* and *ZB* are different. Again, by controlling the values of *ZA* and *ZB* by means of MEMS switches, the amount of energy transfer among modes can be controlled. This transition and other described in [38–40] are the base for building multimodal uniplanar reconfigurable circuits [6, 9–11, 58] using MEMS switches.

**Figure 5.** (a) Impedances connecting the two outer CPW strips. (b) Multimodal circuit model.

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**Figure 6.** (a) Asymmetric shunt impedances connecting the strips of a CPW. (b) Multimodal circuit model.
