2. Description of the principle used to "extract"energy from the vacuum

As a preamble, we hope and suppose that the events which induce the attractive force of Casimir are exerted in a universal, isotropic, perpetual, and immediate way, if the conditions of separation between reflecting Casimir plates are suitable.

Let therefore be a Casimir reflector device consisting of:

Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197


In order for the movement of the movable plate of this Casimir reflector to create electric charges that can be used to induce an attractive Coulomb force, it is necessary that the movement of this movable plate naturally induces a deformation of a structure creating electric charges. A piezoelectric device is therefore required, rigidly connected to the mobile Casimir electrode, so that its induced naturally deformation leads to the appearance of electric charges! And this without any other energy being involved (Figures 3–5).

Of course, it is also necessary that the movement of this Casimir reflector mobile plate Ss2 can be stopped at a chosen and predefined value before the bonding of the surface Ss2 on the fixed surface Ss3 takes place! Otherwise, we just definitively collapse the two reflector plates, and no energy extraction is possible! In addition, it is necessary that the mobile system returns to its initial position (or slightly exceeds it).

As said previously, we can then imagine that the attractive Casimir force exerted between the facing surfaces SS2 and SS3 and which moves the mobile reflector plate SS2, induces a deformation of a parallelepiped piezoelectric bridge of surface SP1 and SP2, rigidly linked, by a metal finger, to this reflecting mirror (Figures 3–5).

We observe, in Figure 5, that the surfaces Sp1 ¼ Sp2 ¼ bp <sup>∗</sup> lp, green or red metallic of the insulating piezoelectric bridge, are connected for:

1. for Sp1 on the face n°1, through the metal finger (green) to the mobile plate of the Casimir surface reflector Ss2 ¼ bs <sup>∗</sup> ls <sup>¼</sup> Ss1 which forms one of the electrodes of the Casimir reflector. Thus, the metallic surfaces Sp1 and the metallic parallelepiped Ss1 ¼ Ss2 are equipotential

2. for Sp2 on the face n°2, at the grids of the switch circuits n°1 and n°2.

The deformations caused by the attractive force of Casimir, then produce fixed electric charges for example Qfn1 <sup>¼</sup> ‐Qfp2, on the faces Sp1 and Sp2 of the insulating piezoelectric bridge. These fixed charges in turn attract, from the immediate environment (mass or earth) to which they are connected by circuits 2, mobile electric charges Q mp1 and Q mn2, respectively. These charges are distributed over the metallized surfaces deposited on the insulating piezoelectric bridge, therefore on Sp2 and the gates of the transistors of circuits 1 and 2 as well as on Sp1, the metal block, the sources of the transistors of circuit 1 and the two coupling capacitors of the electronic transformation circuit (see Figures 3 and 8).

Let SMOS be the surface of the gates of the MOS of switch circuits 1 and 2. The mobile charges, for example positive Q mp2, located on the surface Sp2 of face 2 going to the gate of a MOSNE enriched transistor in the ratio Q mp2MOS <sup>¼</sup> <sup>Q</sup> mp2 <sup>∗</sup> SMOS=Sp2, then produce a positive voltage VG ¼ Q mp2 MOS=COX on the gate of the MOS transistors, with Cox the gate capacitance of the MOS transistors.

Depending on the sign of these mobile charges on the gates of the MOSNE or MOSPE transistors, they can turn one of them ON, if they are sufficient to induce a voltage VG greater than their threshold voltage VTE, positive for the MOSNE transistor and negative on the MOSPE transistor in parallel.

Figure 8. Polarization and applied force and load on a piezoelectric block.

The nature of these charges depends on the initial polarization of the deposited piezoelectric parallelepiped and on the direction of the deformation imposed by the Casimir force. The sign of these mobile charges on the surfaces Sp1 and Sp2 depending on the real polarization obtained during the realization of the piezoelectric material of this bridge, it is the reverse which occurs if the mobile charges are negative on Sp2, hence the parallel setting of switches! (see Figure 3).

As long as this voltage on face 2 of the bridge, for example positive, is less than the threshold voltage VTNE of this MOSNE transistor, the latter remains blocked! Consequently, the mobile charges Qmn1 = �Qmp2 located on the other face Sp1 of the deformed piezoelectric device (connected by a metal block to Ss2) and connected to the sources of the MOSNE and MOSPE remain on these surfaces and do not propagate on the surface of the return electrode. The MOS switches N and P depleted in series from the switch circuit 2 are then on and connect this return electrode to ground (Figure 3).

On the other hand, if this voltage VG becomes greater than the threshold voltage VTNE of the enriched MOSNE, it becomes conducting, and circuit 2 is then blocked, so the mobile charges Q mn1, located on Sp1 and the metal block can cross the MOSNE to homogenize the charge density on all the return electrodes. These electric charges pass through the inductance LIN in series (Figure 3).

When one of the two MOSNE or MOSPE enriched transistors of circuit 1 turns on, then the depleted MOS N and P switches of circuit 2 are blocked. The return electrode, no longer connected to ground, therefore does not discharge these mobile electrical charges and is isolated (Figure 3).

Let Sp1 ¼ Sp2 ¼ lp <sup>∗</sup> bp be the surface area of the faces of the piezoelectric bridge, Sbloc = the surface of the metal block of the Casimir reflector (Figure 5). Let Sr ¼ Sp2 ¼ Sp1 be the surface of the return electrode facing the metallized face Sp2 of the piezoelectric bridge.

The mobiles charges for example negative Q mn1 <sup>¼</sup> ‐<sup>Q</sup> mp2 which was initially distributed on the metallic surfaces Sp1 are distributed, after the closing of the MOSNE switch, on the surfaces Sp1 þ Sr. They induce between the faces Sp2 and Sr, electric charges of opposite sign, an attractive force of Coulomb, parallel and opposite to the attractive force of Casimir.

These same electrical charges opposite on the surfaces Sp2 and Sr. become after distribution as Sp1 = Sr. This charge Q mn1f remains on the return electrode because circuit 2 is blocked (Figures 3 and 5). If the threshold voltages of the transistors are positioned according to VTND ffi VTPE <0< VTNE ffi VTPD, then we have the following configurations depending on the value of the voltage VG.

Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197

As a result, when circuit 1 is blocked, there is no Coulomb electrostatic attraction between Sp2 and Sr. because the metal return electrode Sr. is grounded and therefore free of charges! However, when circuit 1 is on, circuit 2 is then blocked, so as the metallic return electrode Sr. is isolated, the charges of opposite sign and present on the electrodes Sp2 and Sr induce an attractive Coulomb electrostatic force!

This attractive Coulomb force as a first approximation is written:

$$F\_{CO} = \left( S\_S \cdot \frac{\pi^2 \hbar c}{240} \frac{d\_{31} l\_P}{2^\*} \left( \frac{1}{z\_s^4} - \frac{1}{z\_0^4} \right) \right)^2 \left( \frac{1}{4\pi \varepsilon\_0 \varepsilon\_r} \right) \left( \frac{1}{z\_r + z\_0 - z\_S} \right)^2 \tag{5}$$

This attractive force triggered by an input of mobile electric charges of opposite sign on the surfaces Sp2 and Sr is exerted only when the Casimir force between the two reflectors reaches a defined value, dependent on the threshold voltage of the MOSNE transistor.

It will therefore be necessary to adjust the threshold voltage of all these MOSs adequately. Initially, when the piezoelectric beam is not deformed, the electric charges on the faces Sp1,Sp2,Ss2 and Ss3 of the Casimir reflector are zero! The face Ss2 of the Casimir sole plate, distant from z0 from the face Ss3 is then attracted against the fixed face Ss3, only by the force of Casimir. This force is communicated via the connecting finger at the center of the face Sp1 of the piezoelectric bridge and then deforms it.

We will admit in the remainder of this presentation that we are in the case of mode 31 and that, when the piezoelectric bridge undergoes a deformation then the fixed electric charges of the piezoelectric bridge attract negative mobile charges on the electrode Sp1 from the mass and positive mobiles on the other Sp2 electrode.

If, the threshold voltage of the MOSNE is adjusted so that the Coulomb force is triggered only when FCO ¼ p FCA with p proportionality factor 2, then the total repulsion force Ft variable in time and applied to the piezoelectric bridge becomes (Figures 9 and 10).

Figure 9. Piezoelectric bridge cutting reactions and bending moment, deflection.

Figure 10. Final structure with the metal oxides surrounding the metal electrodes.

$$
\overrightarrow{F\_T} = \overrightarrow{F\_{CA}} - \overrightarrow{F\_{CO}} = (1 - p)\overrightarrow{F\_{CA}} \Rightarrow \overrightarrow{F\_T} < 0
$$

Becoming repulsive, this force Ft (dependent on time) induces a deformation of the piezoelectric bridge in the other opposite direction, and the piezoelectric bridge returns or slightly exceeds (because of inertia) its neutral position, without initial deformation, therefore towards its position without any electrical charge.

The variation in time of these mobile charges follows, as a first approximation, a law of distribution of the charges on a short-circuited capacitor. Indeed, the fixed electrode Sr initially at zero potential since at ground, is now isolated by switch circuit 2 which is open and isolates it from ground! This temporal variation of the charges is given by the well-known exponential form of discharge of a capacitor according to the formula:

$$Q\_{mn} = Q\_{mn2} \cdot \exp\left(-\frac{t}{R\_m \mathbf{C}\_S}\right) \tag{6}$$

This variation in mobile charges stops when these electrical charges Q mn are uniformly distributed over the two electrodes Sp2 and Sr. and are equal to Q mn2=2 ¼ ‐<sup>Q</sup> mn1=2 on the two electrodes. Therefore, at time te <sup>¼</sup> RmCs ln(2) (Eq. (5)), te being the time to reach equilibrium, with Rm the ohmic resistance of the metal track Lin of the inductance, Cs the capacitance formed by the electrodes Sp2 and Sr and the input capacitances of the electronics (Figure 11).

This homogenization of electric charges within a metallic conductor:


Figure 11. Interval between casimir electrodes as a function of time for a proportionality coefficient FCO / FCA = 2.

Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197

We therefore obtain a current peak during this homogenization with a duration te of the order of a nanosecond! This current peak IIN circulating for the duration of time te is:

$$I\text{III} = d \frac{Qmn}{d} t''\ I\_{IN} = -\frac{Q\_{mm2}}{R\_m C\_s} \left(\text{Exp}\left(-\frac{t}{R\_m C\_s}\right)\right) \tag{7}$$

We therefore obtain a current peak during this homogenization with a duration te of the order of a nanosecond! A current peak is obtained at time t = 0. With Q mn2/ 2 the charge which is distributed uniformly over the two electrodes Sp2 and Sr. The time t is counted from the closing of one of the transistors of circuit 1 and the opening of the switches of circuit 2. This current peak IIN crossing an inductance LIN during the time te, induces a voltage UIN at the terminals of this inductance LIN as a function of time according to the usual formula:

$$\begin{split} \mathbf{U}\_{IN} &= \mathbf{L}\_{IN} \frac{d(I\_{IN})}{dt} = \mathbf{L}\_{IN} \frac{\mathbf{Q}\_{mm2}}{R\_m \mathbf{C}\_s} \left( \exp \left( -\frac{t}{R\_m \mathbf{C}\_s} \right) \right) \\ &= \mathbf{L}\_{IN} \frac{\mathbf{Q}\_{mm2} \ln(2)}{R\_m \mathbf{C}\_s t\_e} \left( \exp \left( -\frac{t}{R\_m \mathbf{C}\_s} \right) = \mathbf{L}\_{IN} \mathbf{I}\_{IN} \ln \frac{2}{t} e \end{split} \tag{8}$$

There is therefore a voltage peak across the inductance and the electronics appearing without power supply at time t = 0! As the deformations of the piezoelectric bridge cancel each other out during its "rise", the mobile charges on the surfaces Sp1 as well as Sp2 also cancel each other out! As a result, the gate voltage on circuit 1 and 2 MOSs drops below the threshold voltages and circuit 1 blocks. Circuit n°2 turns on again and connects face 2 of return electrode to ground, so the electrical charges on the bridge and the Sr. electrode cancel each other out! (see Figure 5).

The force of Casimir FCA, still present, again attracts the metallic surface SS2 against SS3 and the events described above are repeated. Casimir's force deforms this bridge again and it seems that all starts all over again! The consequence is that the structure made up of the piezoelectric bridge, the connecting finger, the metal block forming the mobile Casimir electrode starts to vibrate, with a frequency dependent:

	- a. of the starting z0 and zr separation interface
	- b. geometric dimensions of the different electrodes,

As we will see, this frequency is lower than that of the first resonant frequency of the mobile structure if the initial interface z0 is not weak enough (< 150 A°) to induce a sufficient Casimir force (see Chapters V and X)! An AC voltage peak UIN is therefore automatically recovered at the terminals of the solenoid LIN. This AC voltage peak can then be rectified to a DC voltage of a few volts, by suitable electronics operating without power supply (see amplification electronics without VI power supply).

Before moving on to theoretical calculations and mathematical simulations of the structure we wish to emphasize that the alternating signal UIN is obtained without the input of any external energy!

In conclusions it seems (except errors) that all the electro-physical phenomena leading to a vibration of the structure and to the production of a voltage modulation are only the consequence of a first phenomenon which is at the origin of the Force of Casimir induced by fluctuations in vacuum energy.

They occur naturally and automatically without the input of any external energy except that of a vacuum … without contradicting Noether's theorem!

#### 3. Calculation of the current generated by the Casimir structure

If the initial separation interface z0 is greater than 150 A°, the forces present are too weak to induce a vibration frequency of the device corresponding to its first resonant frequency (see Chapter V). We sought the numerical solutions of the differential equations obtained and unfortunately insoluble analytically when the device does not vibrate at its first resonant frequency!

#### 3.1 Calculation of the frequency of vibration of the Casimir structure

Let us calculate the evolution in time of the force of Casimir which is applied between the two electrodes separated by an initial distance z0. Apply the theorem of angular momentum to this vibrating structure. The angular momentum of the device is

$$\overrightarrow{\sigma\_{A\_{x,y,x}}^{S}(structure)} = \overbrace{I\_{A\_{x,y,x}}^{S}}^{S} \overrightarrow{\Omega\_{A}^{S}} \tag{9}$$

With the angular momentum vector of the structure, the inertia matrix of the structure with respect to the reference (A, x, y, z) and the rotation vector of the piezoelectric bridge with respect to the axis Ay with α the low angle of rotation along the y axis of the piezoelectric bridge.

$$\begin{aligned} \text{We have because } \mathbf{z} &< \mathbf{l\_p} \sin \left( a \right) = \sin \left( 2 \mathbf{\dot{q}} \right) \approx \frac{2}{l\_P} \overrightarrow{\mathbf{z}} \Rightarrow \overrightarrow{\mathbf{\Omega}\_A^S} \\ \mathbf{z} &= \begin{pmatrix} \mathbf{0} \\ \mathbf{^{da}} / \mathbf{d}t \\ \mathbf{0} \end{pmatrix} \text{ with } \mathbf{\dot{a}} \mathbf{\dot{q}} \approx \frac{2}{l\_P} \overrightarrow{\mathbf{d}t} \end{aligned}$$

Let (Gp, x, y, z), (Gi, x, y, z), (Gs, x, y, z) be the barycentric points respectively of the piezoelectric bridge, of the metal connecting finger and of the metal block constituting the sole mobile of the Casimir reflector. We have (Figure 5):

$$
\overrightarrow{AG\_{P,x,y,x}} = \frac{1}{2} \begin{pmatrix} l\_P \\ b\_P \\ a\_P \end{pmatrix} \quad \overrightarrow{AG\_{I,x,y,x}} = \frac{1}{2} \begin{pmatrix} l\_P & +l\_i \\ b\_P & +b\_i \\ a\_P & +a\_i \end{pmatrix} \quad \overrightarrow{AG\_{S,x,y,x}} = \frac{1}{2} \begin{pmatrix} l\_P & +l\_i + l\_s \\ b\_P & +b\_i + b\_s \\ a\_P & +a\_i + a\_s \end{pmatrix}.
$$

The inertia matrix of the bridge, in the frame of reference (Gp, x, y, z) is

$$
\overline{I\_{GP}^p} = \frac{m\_P}{\mathbf{1}\mathbf{2}} \begin{pmatrix} a\_p^2 + b\_p^2 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & l\_p^2 + b\_p^2 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & l\_p^2 + a\_p^2 \end{pmatrix} \tag{10}
$$

Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197

The inertia matrix of the finger is, in the frame of reference (Gi, x, y, z) is:

$$
\overline{I\_{GI}^l} = \frac{m\_i}{\mathbf{1}\mathbf{2}} \begin{pmatrix} a\_i^2 + b\_i^2 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & l\_i^2 + a\_i^2 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & l\_i^2 + b\_i^2 \end{pmatrix} \tag{11}
$$

The inertia matrix of the reflector is, in the frame of reference (GS, x, y, z) is:

$$
\overline{I\_{\rm GS}^{\rm C}} = \frac{m\_{\rm S}}{12} \begin{pmatrix} a\_{\rm S}^2 + b\_{\rm S}^2 & 0 & 0 \\ 0 & l\_{\rm S}^2 + a\_{\rm S}^2 & 0 \\ 0 & 0 & l\_{\rm S}^2 + b\_{\rm S}^2 \end{pmatrix} \tag{12}
$$

The total inertia of the structure becomes in the reference (A, x, y, z), Is A,x,y z ¼ I P A,x,y,z þ I I A,x,y,z þ I c A,x,y,z with A at the edge of the embedded piezoelectric bridge and IP A,x,y,z,II A,x,y,z,Ic A,x,y,z is the inertia matrix obtained taking Huygens' theorem applied to this structure. The angular momentum theorem applied to the whole structure gives

$$\begin{aligned} \rightarrow \frac{d\left(\sigma\_{A,x,y,z}^{S}\right)}{dt} &= \overrightarrow{I\_{A,x,y,z}^{S}} \rightarrow \frac{d\left(\Omega\_{A}^{S}\right)}{dt} \rightarrow \overrightarrow{\overrightarrow{I\_{A,x,y,z}^{S}}} \frac{2}{I\_{P}} \begin{pmatrix} 0\\0\\d^{2}z/dt^{2}\\0 \end{pmatrix} \\ &= \overrightarrow{\sum\_{A} \text{Moments on the structure}} = \overrightarrow{M\_{A}} + \overrightarrow{M\_{B}} + \overrightarrow{F\_{CA}} \wedge \begin{pmatrix} 2/I\_{P} \\0\\0\\0 \end{pmatrix} \end{aligned} \tag{10}$$
 
$$\text{with } \overrightarrow{F\_{CA}} = \begin{pmatrix} 0\\0\\F\_{CA} \end{pmatrix}$$

We know (see X), according to the axis of Az that the moments: MAY ¼ MBY ¼ ‐FCA lp/8, Therefore: <sup>Ʃ</sup> Moments on the structure relative to the axe Ay <sup>¼</sup> <sup>1</sup>=4<sup>∗</sup> lp<sup>∗</sup> FCA.

Any calculation done

$$\overline{I\_{\mathcal{V}}^{\mathbb{S}}} \frac{2}{l\_P} \frac{d^2 z}{dt^2} = \frac{l\_P}{4} F\_{CA} = \frac{l\_P}{4} \mathcal{S}\_{\mathbb{S}} \frac{\pi^2 \hbar c}{24 \mathbf{0}} \frac{\mathbf{1}}{z^4} \tag{14}$$

with ISy the inertia of the structure relatively to the axe Ay. See Eq. (14) below.

$$\begin{split} \overline{I\_{y}^{S}} &= \rho\_{P} \mathfrak{a}\_{Pbyl \circ} \left( \frac{l\_{P}^{2} + a\_{P}^{2}}{12} + \frac{l\_{P}^{2} + a\_{P}^{2}}{4} \right) + \rho\_{I} \mathfrak{a}\_{ibyl} \left( \frac{l\_{i}^{2} + a\_{i}^{2}}{12} + \frac{(l\_{P} + l\_{i})^{2} + (a\_{P} + a\_{i})^{2}}{4} \right) \\ &+ \rho\_{S} \mathfrak{a}\_{Sbyl} \left( \frac{l\_{S}^{2} + a\_{S}^{2}}{12} + \frac{(l\_{P} + l\_{i} + l\_{S})^{2} + (a\_{P} + a\_{i} + a\_{S})^{2}}{4} \right) \end{split} \tag{15}$$

with ρP,ρI,ρS, respectively the densities of the piezoelectric bridge, the intermediate finger and the mobile electrode of the Casimir reflector.

We then obtain the differential equation which makes it possible to calculate the interval between the two electrodes of the Casimir reflector as a function of time during the "descent" phase when the Coulomb forces are not present.

$$\frac{d^2z}{dt^2} = \frac{l\_P^2}{8} \quad \text{S}\_S \frac{\pi^2 \hbar c}{240} \frac{1}{z^4} = \frac{B}{z^4} \quad \text{with } B = \frac{l\_P^2}{8I\_Y^S} \text{S}\_S \frac{\pi^2 \hbar c}{240} \tag{16}$$

Coulomb forces do not intervene yet because the MOS switches in parallel of circuit 1—before the inductance Lin—are open and the MOS switches in series of circuit 2—after the inductance Lin—being closed, the return Coulomb electrode is to earth. The fixed Casimir electrode is always to earth (see Figures 3 and 5). Coulomb forces will intervene when the gate voltage VG ¼ Q mp2 MOS=COX on the MOSs of circuit n°1 exceeds the threshold voltage of one of them and when circuit n°2 of the depleted N and P MOSs in series will be open (Figures 3 and 5)! Then the switches of the circuit of the parallel MOS transistors will close.

The switches of the series MOS circuit will open and the charge Q mn1 initially present exclusively on the electrode of the bridge and of the metallic block will be distributed uniformly over the second part of coulomb electrodes according to:

$$Q\_{mm1f} = Q\_{mm1} \frac{\mathbf{S}\_{P1}}{\mathbf{S}\_{P1} + \mathbf{S}\_R} \approx \frac{Q\_{mm1}}{2}.4$$

Because Sr ¼ Sp1, Just at the moment of closing circuit n°1 and opening circuit n°2 (Figure 5) we have FCO <sup>¼</sup> ‐p FCA with p a coefficient of proportionality <sup>≥</sup><sup>2</sup> defined by the threshold voltages of the MOS interrupters. The total force FT exerted in the middle of the piezoelectric bridge just at the start of the charge transfer becomes FT <sup>¼</sup> FCA � FCO <sup>¼</sup> FCA � <sup>p</sup><sup>∗</sup> FCA <sup>¼</sup> FCA (1-p).

The "descent" time of the free Casimir electrode will therefore stop when FCO <sup>¼</sup> ‐p FCA.

However, we know that: Fca <sup>¼</sup> <sup>S</sup> <sup>π</sup>2ħ<sup>c</sup> 240z<sup>4</sup> S � �

1.The Casimir force is variable in time and its equation is Eq. (1):

2.The mobile charge on the Casimir electrodes variable also in time (Eq. (3)) is:

$$Qmn2 \approx \frac{Qmn}{2} = \frac{d\_{31}F\_{CA} \, l\_P}{2a\_P}$$

3.The Coulomb force (4), variable over time, acting in opposition to the Casimir force (Eq. (4)):

$$F\_{\rm CD} = pF\_{\rm CA} \Rightarrow \left( S\_{\rm S} \frac{\pi^2 \hbar c}{240} \, \frac{d\_{\rm 3I} l\_P}{2^\ast a\_P} \left( \frac{1}{z\_\circ^4} - \frac{1}{z\_\circ^4} \right) \right)^2 \left( \frac{1}{4\pi\varepsilon\_0 \varepsilon\_r} \right) \left( \frac{1}{z\_r + z\_0 - z\_\circ} \right)^2 \Big|\_{} = pS\_{\rm S} \frac{\pi^2 \hbar c}{240} \, \frac{1}{z\_\circ^4}$$

The differential Eq. (15) unfortunately does not have a literal solution and we programmed on MATLAB the solution of this differential equation "descent"

Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197

and calculated the duration of this "descent" of free Casimir electrode. The duration of the "descent" depending on the desired value of the coefficient of proportionality p, which is regulated by the values of the threshold voltages of the MOS transistors and defined during the manufacture of the device. The "descent" of the free Casimir electrode stops when the inter electrode interface zs is such that:

$$z\_s^4 \left( \left( \frac{1}{z\_r + z\_0 - z\_S} \right)^2 \left( \frac{1}{z\_S^4} - \frac{1}{z\_0^4} \right) \right) = p \frac{3840 \pi e\_0 e\_r}{\pi^2 \hbar c S\_S} \left( \frac{a\_P}{d\_{31} l\_P} \right)^2 \tag{17}$$

This programmable equation gives the time td of the "descent" of the structure submitted to the Casimir force and:

a. depend on the coefficient of proportionality p,

$$F\_T = (\mathbf{1} - p)F\_{CA} \Rightarrow (\mathbf{1} - p)\mathbf{S}\_S \frac{\pi^2 \hbar c}{240} \frac{\mathbf{1}}{z\_{sm}^4} < 0 \quad \text{if } p > 1$$


$$\begin{split} F\_{T} &= F\_{CA} - F\_{CO} \\ &= S\_{\mathbb{S}} \left( \frac{\pi^{2} \hbar}{240} \right) \left( \frac{1}{\varepsilon\_{\text{s}}^{4}} - S\_{\mathbb{S}} \frac{\pi^{2} \hbar}{240} \left( \frac{d\_{3} \cdot 1}{a\_{P}} \right)^{2} \right) \left( \frac{1}{16 \text{ } \pi \text{ } \varepsilon\_{0} \text{ } \varepsilon\_{r}} \right) \left( \frac{1}{\varepsilon\_{\text{s}}^{4}} - \frac{1}{\varepsilon\_{\text{s}}^{4}} \right)^{2} \left( \frac{1}{\left( \text{z}\_{r} + \text{z}\_{0} - \text{z}\_{\text{S}} \right)^{2}} \right) \end{split} \tag{18}$$

The piezoelectric bridge subjected to this force then rises towards its neutral position. The Casimir inter electrode interval increases causing the Casimir force to decrease! As the deformations of the piezoelectric bridge decrease, the electric charge present on the piezoelectric face's decreases, which consequently leads to a drop in the Coulomb Force. The FT force therefore rapidly approaches the starting FCA force, during the "ascent" of the Casimir electrodes.

Let us calculate the duration of this "rise" of the mobile electrode of the Casimir reflector triggered when FCO <sup>¼</sup> <sup>p</sup><sup>∗</sup> FCA.

To know the time taken by the structure to "go back" to its neutral position, we must solve the following differential equation (Eq. (18)):

$$\begin{split} \frac{d^2 \mathbf{z}}{dt^2} &= \frac{l\_P^2}{8J\_Y^8} (F\_{CA} - F\_{CO}) \\ &= \frac{l\_P^2}{8J\_Y^8} \left( lsb\_S \frac{\pi^2 \hbar c}{240} \frac{\mathbf{1}}{z\_s^4} - lsb\_S \frac{\pi^2 \hbar c}{240} \frac{d\_{31}l\_P}{2^\*} \frac{\mathbf{1}}{4\pi e\_0 c\_r} \left( \frac{\mathbf{1}}{z\_s^4} - \frac{1}{z\_0^4} \right)^2 \left( \frac{\mathbf{1}}{z\_r + z\_0 - z\_S} \right)^2 \right) \end{split}$$

By posing <sup>A</sup><sup>1</sup> <sup>¼</sup> lsbS <sup>π</sup>2ħ<sup>c</sup> <sup>240</sup> � � the differential Eq. (17) concerning the "ascent" of the bridge is written:

$$\begin{split} \frac{d^2 \mathbf{z}}{dt^2} &= \frac{l\_P^2}{8. \ I\_Y^S} (F\_{CA} - F\_{CO}) \\ &= \frac{l\_P^2}{8. \ I\_Y^S} A \mathbf{1} \left( \frac{\mathbf{1}}{\mathbf{z}\_s^4} - A \mathbf{1} \left( \frac{d\_3}{a\_P} \right)^2 \right) \left( \frac{\mathbf{1}}{16 \ \pi \ \varepsilon\_0 \ \varepsilon\_r} \right) \left( \frac{\mathbf{1}}{\mathbf{z}\_s^4} - \frac{\mathbf{1}}{\mathbf{z}\_0^4} \right)^2 \left( \frac{\mathbf{1}}{\left( \mathbf{z}\_r + \mathbf{z}\_0 - \mathbf{z}\_S \right)^2} \right) \end{split} \tag{19}$$

This differential Eq. (18) has no analytical solution and can only be solved numerically. We programmed it on MATLAB with the inter-electrode distance zs belonging to the interval [zsm,z0]. The properties and dimensions of the different materials used in this simulation are as follows (Table 1).

The metal used for the Casimir reflector block is Aluminum with a density of 2:7 g cm�3.

In these MATLAB calculations we considered that the metal of the electrodes and metal block was oxidized over a thickness allowing to have an interface between Casimir electrodes of 200 A° (see Chapter 5) which modifies the mass and the inertia of the vibrating structure. It turns out that the choice of aluminum as the metal deposited on these electrodes is preferable given:

1.The ratios between the thickness of the metal oxide obtained and that of the metal attacked by the growth of this oxide during its thermal oxidation (see Chapter V).


#### Table 1.

Table of characteristics used for MATLAB and ANSYS simulations.

Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197

Figure 12. Vibrations of the structure for a coefficient of proportionality p = FCO / FCA = 200: PZT.

2.As its density is weak, we chose aluminum for the purpose to increasing and optimizing the vibration frequency of the structure by minimizing the inertia of the Casimir's reflector and parallelepiped block transferring the Casimir force to the piezoelectric bridge. The mass M of the vibrating structure is then:

$$\mathbf{M} = \mathbf{d\_{pm}}^\* \left( \mathbf{a\_s}^\* \mathbf{b\_s}^\* \mathbf{l\_s} + \mathbf{a\_i}^\* \mathbf{b\_i}^\* \mathbf{l\_i} \right) + \mathbf{d\_{om}}^\* \mathbf{2}^\* \mathbf{z\_{uf}}^\* \left( \mathbf{a\_{so}}^\* \mathbf{b\_{so}} + \mathbf{b\_{so}}^\* \mathbf{l\_{so}} + \mathbf{a\_{so}}^\* \mathbf{l\_{sa}} \right) + \mathbf{d\_p}^\* \left( \mathbf{a\_p}^\* \mathbf{b\_p}^\* \mathbf{l\_p} \right);$$

With dpm the density of the metal, as,bs,ls the geometries of the final metal part of the Casimir electrode sole, dom the density of the metal oxide, aso,bso,lso the geometries of the oxidized parts around the 6 faces of the metal block, dp the density of the piezoelectric parallelepiped (see Figure 12):

#### 3.2 Calculation of the current peak

Let us estimate the duration of the current peak linked to the circulation and homogenization on the return electrodes of the mobile charges. Let Rm be the ohmic resistance of the metals used for the surface electrodes Sp1 + the LIN solenoid + the Sr electrode (see Figure 3) and Cs the capacitance formed by the gap between the return electrodes Sp2 and Sr. Then the current peak circulating during the transition of the mobile loads between Sp1 to SR via circuit n°1 and the LIN solenoid is, as we have seen:

$$I\_{IN} = \frac{Q\_{mn2}}{R\_m C\_S} Exp\left(-\frac{t}{R\_m C\_S}\right).$$

The time t being counted from the closing of the MOSNE switch. The duration of this current peak is estimated at te ¼ Rm:Cs. Log (2), when the charges on each electrode will be Q mn2/2. This current peak is present even if the switch transistors may close some time after, because the mobile charges have already propagated.


current peak (based on an estimate to propagate in a LIN coil of about 10�<sup>5</sup> Henri) gives te ffi <sup>10</sup>�9s. This current peak, passing through a LIN solenoid develops a voltage peak UIN <sup>¼</sup> LINIINP=te <sup>¼</sup> LIN <sup>∗</sup> <sup>Q</sup> mn2<sup>=</sup> <sup>2</sup>:te <sup>∗</sup> ð Þ Rm:Cs which will be exploited by integrated electronics without any power supply described in Chapter IV.

We present below the results of the MATLAB simulations carried out by numerically calculating the differential Eqs. (15) and (18). These numerical calculations give the vibration frequency of the structure which, as we will see, vibrates at a frequency lower than its first resonant frequency (IV).

This vibration frequency depends on the characteristics of the structure (Nature of the piezoelectric material, nature of the metallic conductors, initial interface z0 and zr between Casimir electrodes and return electrodes, geometric dimensions of the Casimir reflectors, coefficient of proportionality p ¼ FCO=FCA … )) (See IV and Annex).
