3. Capacitive RF MEMS mechanical modeling

Before designing any device using RF MEMS technology, it is required to understand the behavior of the main block, the switch. In this way, the first design step will be to compute the applied force a moving beam suffers due to an external electric field. It will be derived considering that the control signal is an electric potential applied between the movable beam and an activation electrode.

The uniform bridge of MEMS is geometrically simple since it is only a rectangular suspension located above a tape connected to contact pads by the sides of the same width, as shown in

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power…

The mechanical model of deformable flat capacity is given by Eq. (1), where g is the height between the low beam and the electrode. The bridge width is denoted by Wb and the length of

The height depends on the voltage applied between the electrodes. In the absence of voltage

dWt ¼ dWe � dWm ¼ ð Þ� V:dq Fe:dge

where q is the quantity of charge accumulated in the capacity and Fe is the electrostatic force.

The well-known equation of potential of electrostatic energy is given by Eq. (3):

Then, the electrostatic force for a flat capacity can be expressed by Eq. (4):

Fe <sup>¼</sup> <sup>∂</sup>We

<sup>U</sup>EðÞ¼ <sup>j</sup> <sup>1</sup> 2

<sup>∂</sup><sup>g</sup> ¼ � <sup>1</sup>

Fm ¼ �Fe <sup>¼</sup> kz <sup>g</sup><sup>0</sup> � <sup>g</sup> ¼ � <sup>1</sup>

<sup>2</sup> <sup>ε</sup><sup>0</sup>

The mechanical behavior of the beam can be modeled by a spring of constant kz. This induces a mechanical force (Fm) exerted by the bridge. This force is the opposite of Fe and it is defined

WbW

<sup>2</sup> <sup>ε</sup><sup>0</sup>

WbW

WbW

<sup>g</sup> (1)

C Vð Þ:V<sup>2</sup> (3)

<sup>g</sup><sup>2</sup> <sup>V</sup><sup>2</sup> (4)

<sup>g</sup><sup>2</sup> <sup>V</sup><sup>2</sup> (5)

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99

C ¼ ε<sup>0</sup>

Nominal capacity: the capacitance C between the two electrodes is given by:

(V = 0), the height is equal to g0 and the capacity is named C0 [8]:

Figure 3 [18].

by Eq. (5):

Figure 3. The beam of the MEMS under analysis.

the ground electrode by W:

#### 3.1. Capacitive RF MEMS device configuration

The performance, as well the analysis used to improve the RF MEMS device, is heavily dependent on the study of the bridge. Figure 2 presents the proposed RF MEMS switch, with small dimensions (1200 <sup>900</sup> <sup>681</sup> <sup>μ</sup>m<sup>3</sup> ). This RF MEMS is based on CPW technology (G/S/G) = (90/120/90).

This RF MEMS varactor structure has a multilayer configuration. The used substrate is based on silicon (Si) with thickness of 675 μm. The second layer is made of a silicon dioxide (SiO2) with thickness equal to 2 μm and a CPW line circuit metal based on copper with thickness of 1 μm. The bridge has a depth of 1 μm, with ends attached to the groundline of the CPW by an epoxy (polymer based on negative-tone photoresist SU-8 2000.5 with 3 μm thickness). The dielectric is fabricated through a silicon nitride (Si3N4) with depth equal to 1 μm.

#### 3.2. Mechanical model

The first step when modeling a RF MEMS device is to determine the electromechanical behavior of the switch, meaning that we want to understand how, and how much, the structure will move in response to an applied voltage.

Figure 2. Design of the proposed capacitive RF MEMS device.

The uniform bridge of MEMS is geometrically simple since it is only a rectangular suspension located above a tape connected to contact pads by the sides of the same width, as shown in Figure 3 [18].

Nominal capacity: the capacitance C between the two electrodes is given by:

3. Capacitive RF MEMS mechanical modeling

3.1. Capacitive RF MEMS device configuration

with small dimensions (1200 <sup>900</sup> <sup>681</sup> <sup>μ</sup>m<sup>3</sup>

ture will move in response to an applied voltage.

Figure 2. Design of the proposed capacitive RF MEMS device.

and an activation electrode.

98 MEMS Sensors - Design and Application

(G/S/G) = (90/120/90).

3.2. Mechanical model

Before designing any device using RF MEMS technology, it is required to understand the behavior of the main block, the switch. In this way, the first design step will be to compute the applied force a moving beam suffers due to an external electric field. It will be derived considering that the control signal is an electric potential applied between the movable beam

The performance, as well the analysis used to improve the RF MEMS device, is heavily dependent on the study of the bridge. Figure 2 presents the proposed RF MEMS switch,

This RF MEMS varactor structure has a multilayer configuration. The used substrate is based on silicon (Si) with thickness of 675 μm. The second layer is made of a silicon dioxide (SiO2) with thickness equal to 2 μm and a CPW line circuit metal based on copper with thickness of 1 μm. The bridge has a depth of 1 μm, with ends attached to the groundline of the CPW by an epoxy (polymer based on negative-tone photoresist SU-8 2000.5 with 3 μm thickness). The

The first step when modeling a RF MEMS device is to determine the electromechanical behavior of the switch, meaning that we want to understand how, and how much, the struc-

dielectric is fabricated through a silicon nitride (Si3N4) with depth equal to 1 μm.

). This RF MEMS is based on CPW technology

The mechanical model of deformable flat capacity is given by Eq. (1), where g is the height between the low beam and the electrode. The bridge width is denoted by Wb and the length of the ground electrode by W:

$$\mathbf{C} = \varepsilon\_0 \frac{\mathbf{W}\_b \mathbf{W}}{\mathbf{g}} \tag{1}$$

The height depends on the voltage applied between the electrodes. In the absence of voltage (V = 0), the height is equal to g0 and the capacity is named C0 [8]:

$$dW\_t = dW\_\varepsilon - dW\_m = (V.d\boldsymbol{q}) - \text{(Fe.d\mathbf{g}\_\varepsilon)}\tag{2}$$

where q is the quantity of charge accumulated in the capacity and Fe is the electrostatic force.

The well-known equation of potential of electrostatic energy is given by Eq. (3):

$$\mathbf{U}\_E(j) = \frac{1}{2}\mathbf{C}(V).V^2\tag{3}$$

Then, the electrostatic force for a flat capacity can be expressed by Eq. (4):

$$F\_e = \frac{\partial W e}{\partial \mathbf{g}} = -\frac{1}{2} \left. \varepsilon\_0 \frac{W\_b W}{\mathbf{g}^2} V^2 \right. \tag{4}$$

The mechanical behavior of the beam can be modeled by a spring of constant kz. This induces a mechanical force (Fm) exerted by the bridge. This force is the opposite of Fe and it is defined by Eq. (5):

$$F\_m = -F\_\varepsilon = k\_z(\mathbf{g}\_0 - \mathbf{g}) = -\frac{1}{2} \,\varepsilon\_0 \frac{W\_b W}{\mathcal{g}^2} V^2 \tag{5}$$

Figure 3. The beam of the MEMS under analysis.

From this equation, we can then conclude a relationship between the height of the air gap g and the applied voltage V:

$$0 = \mathbf{g}^3 - \mathbf{g}\_0 \mathbf{g}^2 + \frac{1}{2k\_m} \varepsilon\_0 W\_b W V^2 \tag{6}$$

The relationship between the applied voltage and the spacing g parameter is given in Eq. (7):

$$V\_p = \sqrt{\frac{2k\_z}{\varepsilon\_0 W\_b W} g^2 (\mathbf{g}\_0 - \mathbf{g})} \tag{7}$$

$$k\_z = \frac{1}{2} \left( 32Ew \left( \frac{t}{l} \right)^3 + 8\sigma (1 - \mathfrak{d})w \left( \frac{t}{l} \right) \right). \tag{8}$$

where E is the Young's modulus, σ is the residual stress of the beam, υ is the Poisson's coefficient, t is the thickness, and l is the length of the bridge.

In Table 3, the mechanical properties (Young's modulus, Poisson's ratio, residual stress, and density) of four different bridge materials (nickel, copper, aluminum, and gold) are presented.

In terms of control voltage, comparatively the nickel presents a bad choice, since the required voltage to obtain some deflection is near twice the voltage that is required for gold and aluminum. However, the gold price presents an obstruction, being the best choice, in this comparative study, the aluminum.

The bridge was simulated using COMSOL multiphysic and reaches a deflection equal to 2 μm. The obtained simulation results are given in Figure 4 for applied voltage equal to 25 V. The relationship between the capacitance and the applying voltage is shown in Figure 5.

#### 3.3. RF MEMS switch design parameters

We will present next the relevant parameters that define the variable MEMS capacity.

Tuning range: the "tuning range" or variation of the capacity is an important factor of the variance MEMS capacities. It is defined as


TR <sup>¼</sup> Cmax � Cmin Cmin

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Quality factor: the quality factor Q of a component is an important parameter. Indeed, it determines the losses of a variable filter or the noise of a VCO using a variable capacity. It is

defined by the ratio between the energy stored and the energy lost by the component:

Figure 5. Relationship between the capacity and the voltage applied.

Figure 4. Simulation results of the beam.

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Table 3. Mechanical parameters of different materials of the bridge.

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power… http://dx.doi.org/10.5772/intechopen.74785 101

Figure 4. Simulation results of the beam.

From this equation, we can then conclude a relationship between the height of the air gap g

The relationship between the applied voltage and the spacing g parameter is given in Eq. (7):

2kz

1 2km

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>ε</sup>0WbW <sup>g</sup><sup>2</sup> <sup>g</sup><sup>0</sup> � <sup>g</sup> � �

! � �

where E is the Young's modulus, σ is the residual stress of the beam, υ is the Poisson's

In Table 3, the mechanical properties (Young's modulus, Poisson's ratio, residual stress, and density) of four different bridge materials (nickel, copper, aluminum, and gold) are presented. In terms of control voltage, comparatively the nickel presents a bad choice, since the required voltage to obtain some deflection is near twice the voltage that is required for gold and aluminum. However, the gold price presents an obstruction, being the best choice, in this

The bridge was simulated using COMSOL multiphysic and reaches a deflection equal to 2 μm. The obtained simulation results are given in Figure 4 for applied voltage equal to 25 V. The

Tuning range: the "tuning range" or variation of the capacity is an important factor of the

Material Nickel Copper Aluminum Gold Young's modulus E (GPa) 200 120 69 79 Poisson's ratio 0.31 0.355 0.345 0.42 Residual stress σ (MPa) 20 20 20 20

Pull-down voltage simulated result [V] 46 34 25 22

) 8900 8960 2700 19,300

relationship between the capacitance and the applying voltage is shown in Figure 5.

We will present next the relevant parameters that define the variable MEMS capacity.

þ 8σð Þ 1 � ϑ w

t l

ε0WbWV<sup>2</sup> (6)

: (8)

(7)

<sup>0</sup> <sup>¼</sup> <sup>g</sup><sup>3</sup> � <sup>g</sup>0g<sup>2</sup> <sup>þ</sup>

s

Vp ¼

<sup>2</sup> <sup>32</sup>Ew <sup>t</sup>

l � �<sup>3</sup>

kz <sup>¼</sup> <sup>1</sup>

coefficient, t is the thickness, and l is the length of the bridge.

and the applied voltage V:

100 MEMS Sensors - Design and Application

comparative study, the aluminum.

3.3. RF MEMS switch design parameters

variance MEMS capacities. It is defined as

Table 3. Mechanical parameters of different materials of the bridge.

Density (kg/m<sup>3</sup>

Figure 5. Relationship between the capacity and the voltage applied.

$$TR = \frac{\mathbf{C\_{max}} - \mathbf{C\_{min}}}{\mathbf{C\_{min}}} \tag{9}$$

Quality factor: the quality factor Q of a component is an important parameter. Indeed, it determines the losses of a variable filter or the noise of a VCO using a variable capacity. It is defined by the ratio between the energy stored and the energy lost by the component:

$$Q = \frac{\text{Loss energy per cycle}}{\text{Total energy per cycle}} \tag{10}$$

A B

known, the next step is to determine each subsystem matrix.

4.1.1. Coplanar waveguide modeling

The ratio K kð Þ=K<sup>0</sup>

presented in Eq. (17):

C D � � <sup>¼</sup> <sup>A</sup><sup>1</sup> <sup>B</sup><sup>1</sup>

C<sup>1</sup> D<sup>1</sup> � �

<sup>∗</sup> Ab Bb Cb Db � �

Considering a system represented by an association of three devices where the ABCD matrix is

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power…

The CPW's most important electrical parameters are the characteristic impedance Zc, and the effective permittivity εeff, both given by Eq. (12), where K (k) and K<sup>0</sup> (k) present the elliptic integral which essentially depends on the CPW's geometric and physical characteristics. Here, ε<sup>r</sup> is the relative permittivity, w is the width of the RF line, s is the gap between the RF line and

> K0 ð Þ k Kð Þ k

ε<sup>r</sup> � 1 2

sh <sup>π</sup><sup>w</sup> 4h � �

sh <sup>π</sup>ð Þ <sup>w</sup> <sup>þ</sup> <sup>2</sup><sup>s</sup> 2h � �

> K kð Þ K0

> K kð Þ K0

The second parameter for modeling the CPW line is the propagation constant. This parameter is given by Eq. (16), as a function of the attenuation constant and the phase constant. The attenuation constant is due to the conductor as well as to the attenuation in dielectric, both

ð Þ <sup>k</sup> <sup>¼</sup> <sup>1</sup> π

ð Þ <sup>k</sup> <sup>¼</sup> <sup>π</sup>

ln 2 <sup>1</sup> <sup>þ</sup> ffiffi

<sup>1</sup> � ffiffi k p !

ln 2 <sup>1</sup> <sup>þ</sup> ffiffi

<sup>1</sup> � ffiffi k p !

k p

> k p

ð Þ¼ <sup>k</sup> <sup>K</sup> <sup>k</sup><sup>0</sup> � � <sup>k</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>k</sup> <sup>p</sup>

> 1 ffiffiffi 2 p )

K0

ð Þ k Kð Þ k1 Kð Þ k K<sup>0</sup>

ð Þ k1


<sup>∗</sup> <sup>A</sup><sup>2</sup> <sup>B</sup><sup>2</sup> C<sup>2</sup> D<sup>2</sup> � �


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(11)

103

(12)

(13)

(14)

(15)


ground, h is the thickness of substrate, and h1 is the thickness of the buffer layer:

ZC <sup>¼</sup> <sup>30</sup><sup>π</sup> ffiffiffiffiffiffi εeff p

8 >>><

>>>:

8 >>>>><

>>>>>:

(

8 >>>>>>><

>>>>>>>:

K kð Þ K0 ð Þ <sup>k</sup> <sup>¼</sup> εeff ¼ 1 þ

<sup>k</sup> <sup>¼</sup> <sup>w</sup> w þ 2s

k<sup>1</sup> ¼

K0

ð Þ k approximation is done by Eq. (15):

0 ≤ k ≤

1 ffiffiffi 2 p ≤ k ≤ 1 )

Linearity: the nonlinearity of the passive components is an important and demanding data for radio-frequency applications. In fact, we want to obtain the value of the linear capacity as a function of the frequency and the actuating voltage.

#### 4. Capacitive RF MEMS electrical modeling

Once the mechanical response to the control voltage is known, the next step will be to model the RF load that the switch will present to a transmission line. That load will be the variable that can be controlled to obtain RF tunable devices or systems.

#### 4.1. RF modeling approach

Figure 6 shows the proposed model of the proposed RF MEMS [19]. The proposed circuit model consists of two CPW lines, separated by a shunt RLC circuit. The RLC is the equivalent bridge circuit.

The MEMS can be modeled by the association of three subsystems in cascade. The ABCD matrix is given by Eq. (11):

Figure 6. Electromagnetics model.

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power… http://dx.doi.org/10.5772/intechopen.74785 103

$$
\begin{bmatrix} A & B \\ \mathbb{C} & D \end{bmatrix} = \underbrace{\begin{bmatrix} A\_1 & B\_1 \\ \mathbb{C}\_1 & D\_1 \end{bmatrix}}\_{\text{TL1}} \* \underbrace{\begin{bmatrix} A\_b & B\_b \\ \mathbb{C}\_b & D\_b \end{bmatrix}}\_{\text{Bridgc}} \* \underbrace{\begin{bmatrix} A\_2 & B\_2 \\ \mathbb{C}\_2 & D\_2 \end{bmatrix}}\_{\text{TL2}} \tag{11}
$$

Considering a system represented by an association of three devices where the ABCD matrix is known, the next step is to determine each subsystem matrix.

#### 4.1.1. Coplanar waveguide modeling

<sup>Q</sup> <sup>¼</sup> Lost energy per cycle

Linearity: the nonlinearity of the passive components is an important and demanding data for radio-frequency applications. In fact, we want to obtain the value of the linear capacity as a

Once the mechanical response to the control voltage is known, the next step will be to model the RF load that the switch will present to a transmission line. That load will be the variable

Figure 6 shows the proposed model of the proposed RF MEMS [19]. The proposed circuit model consists of two CPW lines, separated by a shunt RLC circuit. The RLC is the equivalent

The MEMS can be modeled by the association of three subsystems in cascade. The ABCD

function of the frequency and the actuating voltage.

4. Capacitive RF MEMS electrical modeling

4.1. RF modeling approach

102 MEMS Sensors - Design and Application

matrix is given by Eq. (11):

Figure 6. Electromagnetics model.

bridge circuit.

that can be controlled to obtain RF tunable devices or systems.

Total energy per cycle (10)

The CPW's most important electrical parameters are the characteristic impedance Zc, and the effective permittivity εeff, both given by Eq. (12), where K (k) and K<sup>0</sup> (k) present the elliptic integral which essentially depends on the CPW's geometric and physical characteristics. Here, ε<sup>r</sup> is the relative permittivity, w is the width of the RF line, s is the gap between the RF line and ground, h is the thickness of substrate, and h1 is the thickness of the buffer layer:

$$\begin{cases} Z\_{\mathcal{C}} = \frac{30\pi}{\sqrt{\varepsilon\_{\ell\mathcal{f}}}} \frac{K'(\mathbf{k})}{K(\mathbf{k})} \\\\ \varepsilon\_{\ell\mathcal{f}} = 1 + \frac{\varepsilon\_r - 1}{2} \frac{K'(\mathbf{k})K(\mathbf{k\_1})}{K(\mathbf{k})K'(\mathbf{k\_1})} \end{cases} \tag{12}$$

$$\begin{cases} k = \frac{w}{w + 2s} \\ k\_1 = \frac{sh\left(\frac{\pi w}{4h}\right)}{sh\left(\frac{\pi (w + 2s)}{2h}\right)} \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{cases} \tag{13}$$

$$\begin{cases} \mathbf{k}'(\mathbf{k}) = \mathbf{k}(\mathbf{k}')\\ \mathbf{k}' = \sqrt{1-k} \end{cases} \tag{14}$$

The ratio K kð Þ=K<sup>0</sup> ð Þ k approximation is done by Eq. (15):

$$\frac{K(k)}{K'(\mathbf{k})} = \begin{cases} 0 \le k \le \frac{1}{\sqrt{2}} \Rightarrow \frac{K(k)}{K'(\mathbf{k})} = \frac{\pi}{\ln\left(2\frac{1+\sqrt{k}}{1-\sqrt{k}}\right)}\\\\ \frac{1}{\sqrt{2}} \le k \le 1 \Rightarrow \frac{K(k)}{K'(\mathbf{k})} = \frac{1}{\pi}\ln\left(2\frac{1+\sqrt{k}}{1-\sqrt{k}}\right) \end{cases} \tag{15}$$

The second parameter for modeling the CPW line is the propagation constant. This parameter is given by Eq. (16), as a function of the attenuation constant and the phase constant. The attenuation constant is due to the conductor as well as to the attenuation in dielectric, both presented in Eq. (17):

$$\gamma = a + j\beta \begin{cases} a = a\_c + \alpha\_d \\ \beta = \frac{2\pi}{\lambda}; \lambda = \frac{c}{f\sqrt{\varepsilon\_{eff}\mu\_{eff}}}; c = 3 \times 10^8 \text{ m s}^{-1} \end{cases} \tag{16}$$

A B C D � � <sup>¼</sup>

parameters (Lb), (Rb), and (Cb) are given by Eqs. (19) and (20):

Rb <sup>¼</sup> <sup>L</sup> σwt

8 < :

loss, (c) phase of S11, (d) phase of S12.

4.1.2. Bridge modeling

coshð Þ γL Zcsinhð Þ γL

coshð Þ γL

3

5 (18)

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(19)

105

sinhð Þ γL Zc

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power…

The bridge is modeled by an inductor (Lb) and a resistance (Rb) in series. These parameters are independent of the substrate. The proposed RF MEMS has a capacitive variable (Cb). The

Lb <sup>¼</sup> <sup>0</sup>:2<sup>∗</sup> <sup>ð</sup>ln Lð Þ <sup>=</sup>ð Þ <sup>w</sup> <sup>þ</sup> <sup>t</sup> Þ þ <sup>1</sup>:<sup>193</sup> <sup>þ</sup> <sup>0</sup>:2235<sup>∗</sup> ½ � ð Þ ð Þ <sup>w</sup> <sup>þ</sup> <sup>t</sup> <sup>=</sup><sup>L</sup> <sup>∗</sup><sup>L</sup>

Figure 8. Simulation results of the scattering parameters versus frequency at the ON state: (a) return loss, (b) insertion

2 4

$$\begin{cases} \alpha\_{d} = 27.3 \frac{\varepsilon\_{r}}{\varepsilon\_{r} - 1} \frac{\varepsilon\_{\varepsilon \mathcal{f}} (f) - 1}{\sqrt{\varepsilon\_{\varepsilon \mathcal{f}} (f)}} \frac{\tan \delta}{\lambda\_{0}} (\text{dB/length}) \\\\ \alpha\_{c} = \frac{8.68 R\_{s}}{480 \pi K (k\_{1}) K (k\_{1}^{'}) \left(1 - k\_{1}^{2}\right)} \begin{bmatrix} \frac{1}{a} \left(\pi + Ln \left(\frac{8a\pi (1 - k\_{1})}{t(1 + k\_{1})}\right)\right) \\\\ + \frac{1}{b} \left(\pi + Ln \left(\frac{8b\pi (1 - k\_{1})}{t(1 + k\_{1})}\right)\right) \end{bmatrix} \text{(dB/length)} \end{cases} \tag{17}$$

The model of CPW line is presented by ABCD matrix in Eq. (18):

Figure 7. Simulation results of the scattering parameters versus frequency at the OFF state: (a) return loss, (b) insertion loss, (c) phase of S11, (d) phase of S12.

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power… http://dx.doi.org/10.5772/intechopen.74785 105

$$
\begin{bmatrix} A & B \\ \mathbf{C} & D \end{bmatrix} = \begin{bmatrix} \cosh(\gamma \mathbf{L}) & \mathbf{Z}\_c \sinh(\gamma \mathbf{L}) \\ \sinh(\gamma \mathbf{L}) & \cosh(\gamma \mathbf{L}) \end{bmatrix} \tag{18}
$$

#### 4.1.2. Bridge modeling

γ ¼ α þ jβ

ε<sup>r</sup> � 1

<sup>α</sup><sup>c</sup> <sup>¼</sup> <sup>8</sup>:68Rs 480πK kð Þ<sup>1</sup> K k<sup>0</sup>

<sup>α</sup><sup>d</sup> <sup>¼</sup> <sup>27</sup>:<sup>3</sup> <sup>ε</sup><sup>r</sup>

8

104 MEMS Sensors - Design and Application

>>>>>>>><

>>>>>>>>:

loss, (c) phase of S11, (d) phase of S12.

α ¼ α<sup>c</sup> þ α<sup>d</sup> <sup>β</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

> tanδ λ0

2 1 � �

8 < :

<sup>ε</sup>effð Þ� <sup>f</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>ε</sup>effð Þ<sup>f</sup> <sup>p</sup>

> 1 � � <sup>1</sup> � <sup>k</sup>

The model of CPW line is presented by ABCD matrix in Eq. (18):

<sup>λ</sup> ; <sup>λ</sup> <sup>¼</sup> <sup>c</sup>

1 a

þ 1

Figure 7. Simulation results of the scattering parameters versus frequency at the OFF state: (a) return loss, (b) insertion

f ffiffiffiffiffiffiffiffiffiffiffiffiffi εeff μeff

ð Þ dB=lenght

<sup>p</sup> ; c <sup>¼</sup> <sup>3</sup> � <sup>108</sup> m s�<sup>1</sup>

tð Þ 1 þ k<sup>1</sup>

� � � �

tð Þ 1 þ k<sup>1</sup>

ð Þ dB=lenght

<sup>π</sup> <sup>þ</sup> Ln <sup>8</sup>aπð Þ <sup>1</sup> � <sup>k</sup><sup>1</sup>

� � � �

<sup>b</sup> <sup>π</sup> <sup>þ</sup> Ln <sup>8</sup>bπð Þ <sup>1</sup> � <sup>k</sup><sup>1</sup>

(16)

(17)

The bridge is modeled by an inductor (Lb) and a resistance (Rb) in series. These parameters are independent of the substrate. The proposed RF MEMS has a capacitive variable (Cb). The parameters (Lb), (Rb), and (Cb) are given by Eqs. (19) and (20):

$$\begin{cases} R\_b = \frac{L}{\sigma wt} \\ L\_b = 0.2^\circ [(\ln(L/(\text{w} + \text{t}))) + 1.193 + 0.2235^\circ ((\text{w} + \text{t})/L)] \ast \text{L} \end{cases} \tag{19}$$

Figure 8. Simulation results of the scattering parameters versus frequency at the ON state: (a) return loss, (b) insertion loss, (c) phase of S11, (d) phase of S12.

$$c\_b = \begin{cases} c\_{\text{down}} = \frac{\varepsilon\_0 \varepsilon\_I A}{t \hbar} \\ c\_{up} = \frac{\varepsilon\_0 A}{g\_0 + \frac{t \hbar}{\varepsilon\_I}} \end{cases} \tag{20}$$

To validate our model, we simulated the capacitive RF MEMS switch with HFSS. The results are compared in terms of return loss, insertion loss, and phase at the two states of the capacitive RF MEMS switch (ON and OFF). Here, we intend to show the similarity between

Study and Design of Reconfigurable Wireless and Radio-Frequency Components Based on RF MEMS for Low-Power…

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107

This section will present the use of RF MEMS switches to obtain different tunable RF devices,

The reflection-type phase shifter as shown in Figure 9 is constituted with hybrid coupler and RF MEMS capacitive switches (i.e., metal-dielectric-metal) [20]. The first RF MEMS is connected between a through-port and ground. The second is linked between the coupled port and ground. In this design, the tunability is achieved by the use of a capacitive RF MEMS switch acting as a reflection load. The capacitor value, which is controlled by a DC voltage, operates from downstate to upstate. This variable capacitance is used to tune the variable phase shifter.

The reflection-type phase shifter using switch RF MEMS capacitive was implemented in ADS simulation software. The reflection coefficient Γ is given by Eq. (25), where (Xb ¼ Lbω�

Figure 10 shows the RF phase shifter performance around 18 GHz, in terms of the return and insertion loss, and the phase shift dependency on the applied voltage. Despite not fully linear, it is possible to observe an almost linear characteristic of the phase shifter in different fre-

<sup>c</sup> þ 2ZcRb � ZcZ0 � Z0Rb

<sup>c</sup> þ 2ZcRb þ ZcZ0 þ Z0Rb

<sup>þ</sup> <sup>j</sup>Xb½ � 2Zc � Z0

(25)

<sup>þ</sup> <sup>j</sup>Xb½ � 2Zc <sup>þ</sup> Z0

the results of our model and the software HFSS simulation.

namely, a phase shifter, a resonator, and a tunable antenna.

<sup>Γ</sup> <sup>¼</sup> <sup>∣</sup>Γ∣e<sup>j</sup>Φ<sup>21</sup> <sup>¼</sup> Zs � Z0

Zs þ Z0

Figure 9. Simulation results of the scattering parameters versus frequency at ON state.

<sup>¼</sup> <sup>Z</sup><sup>2</sup>

Z2

ð Þ 1=cbω ).

quency ranges.

5. RF MEMS-based reconfigurable component design

5.1. Reconfigurable phase shifter at 18 GHz based on RF MEMS

The bridge model can be presented by the next ABCD matrix:

$$
\begin{bmatrix} A\_b & B\_b \\ C\_b & D\_b \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \overline{Z\_b} & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \overline{R\_b + jX\_b} & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \overline{R\_b + j(L\_b\omega + 1/\mathbb{C}\_b\omega)} & 1 \end{bmatrix} \tag{21}
$$

#### 4.2. The scattering parameters model

The standard output of simulation tools is the S-parameters. In this way, we need to transform the previous developed model to present it in a convenient way to allow comparison with such tools. The scattering parameters can be expressed by the following form:

$$
\begin{bmatrix} S\_{11} & S\_{12} \\ S\_{21} & S\_{22} \end{bmatrix} = \begin{bmatrix} \frac{A + B/Z\_0 - CZ\_0 - D}{A + B/Z\_0 + CZ\_0 + D} & \frac{2(AD - BC)}{A + B/Z\_0 + CZ\_0 + D} \\\\ \frac{2}{A + B/Z\_0 + CZ\_0 + D} & \frac{-A + B/Z\_0 - CZ\_0 + D}{A + B/Z\_0 + CZ\_0 + D} \end{bmatrix} \tag{22}
$$

where the reflection coefficient and its phase are given by Eq. (23) and the insertion loss and its phase are given by Eq. (24). The scattering parameters are written in the following form:

<sup>S</sup><sup>11</sup> <sup>¼</sup> <sup>S</sup><sup>22</sup> <sup>¼</sup> <sup>∣</sup>S11∣e<sup>j</sup>Φ<sup>11</sup> <sup>¼</sup> Zc <sup>2</sup> � <sup>Z</sup><sup>2</sup> 0 � � <sup>þ</sup> 2RbZcÞ þ j2ZcXb ð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> <sup>þ</sup> 2Rbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc � � <sup>þ</sup> j2Xbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc ¼> ∣S11∣ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zc <sup>2</sup> � <sup>Z</sup><sup>2</sup> 0 � � <sup>þ</sup> 2RbZc � �<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>ZcXb <sup>2</sup> q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> <sup>þ</sup> 2Rbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc � �<sup>2</sup> <sup>þ</sup> ð Þ 2Xbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> r 8 >>< >>: 9 >>= >>; <sup>Φ</sup><sup>12</sup> <sup>¼</sup> tg �<sup>1</sup> <sup>2</sup>ZcXb 2RbZc � � � tg �<sup>1</sup> 2Xbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc ð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> <sup>þ</sup> 2Rbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc � � 0 @ 1 A 8 >>>>>>>>>>< >>>>>>>>>>: (23) <sup>S</sup><sup>12</sup> <sup>¼</sup> <sup>S</sup><sup>21</sup> <sup>¼</sup> <sup>∣</sup>S12∣e<sup>j</sup>Φ<sup>12</sup> <sup>¼</sup> 2RbZc <sup>þ</sup> j2ZcXb ð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> <sup>þ</sup> 2Rbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc � � <sup>þ</sup> j2Xbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc ¼> ∣S12∣ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 2RbZc <sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>ZcXb <sup>2</sup> q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> <sup>þ</sup> 2Rbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc � �<sup>2</sup> <sup>þ</sup> ð Þ 2Xbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> r <sup>Φ</sup><sup>12</sup> <sup>¼</sup> tg �<sup>1</sup> <sup>2</sup>ZcXb 2RbZc � � � tg �<sup>1</sup> 2Xbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc ð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc <sup>2</sup> <sup>þ</sup> 2Rbð Þ <sup>Z</sup><sup>0</sup> <sup>þ</sup> Zc � � 0 @ 1 A 8 >>>>>>>>>< >>>>>>>>>: (24)

The simulation results of the capacitive RF MEMS switch at the two states OFF (the bridge at downstate) and ON (bridge position g = 3 μm) are shown, respectively, in Figures 7 and 8.

To validate our model, we simulated the capacitive RF MEMS switch with HFSS. The results are compared in terms of return loss, insertion loss, and phase at the two states of the capacitive RF MEMS switch (ON and OFF). Here, we intend to show the similarity between the results of our model and the software HFSS simulation.
