8. Energy balance


A quick order of magnitude calculation gives: ΔQvib = heat transmitted by the vibrations of the piezoelectric bridge and evacuated outside follow, in first approximation, the well-known formula ΔQvib ¼ ΔSΔTΔQvib ¼ ΔSΔT, with ΔS entropy variation (J°K�<sup>1</sup> ) and ΔT = temperature variation (°K).

However, we know that: (cf: M. BARTHES, M. Colas des Francs SOLID MECHANICAL VIBRATIONAL PHYSICS, ESTP: (Special School of Public Works).

<sup>Δ</sup>Qvib <sup>¼</sup> <sup>1</sup>=<sup>2</sup> <sup>∗</sup> m 2ð Þ <sup>π</sup>fvib 2 x max <sup>2</sup> ; with: fvib = Vibration frequencies of the piezoelectric bridge, m = mass of this bridge, x max = maximum deflection of the bridge.

This heat, expended at the level of the piezoelectric bridge, causes a temperature increase.

As a first approximation we can say: ΔQvib ¼ mCpiezoΔT, with Cpiezo = Specific heat capacity of the piezoelectric bridge (J Kg�<sup>1</sup> °K�<sup>1</sup> ), ΔT ¼ T emperature variation (°K).

Consequently ΔT ¼ 2ð Þ π fvib 2 x max <sup>2</sup>=Cpiezo = Temperature variation.

We have for example:

Cpiezo <sup>¼</sup> CPMN‐PT <sup>¼</sup> 310 JKg�1°K�<sup>1</sup> � �,fvib ffi <sup>106</sup> Hz,x max ffi <sup>100</sup><sup>∗</sup> <sup>10</sup>�<sup>10</sup> m, we obtain: <sup>Δ</sup><sup>T</sup> ffi <sup>10</sup>�3°K which is negligible!!

So, the expulsion of entropy from the vibrating Casimir Electrode is negligible! From 0 to ze:

$$\text{In a cycle from 0 to z, the energy } E\_{\text{Casimir is:}} \\ \text{is:} \\ E\_{\text{Casimir}} = \int\_0^{\text{xe}} F\_{cd} d\mathbf{z} = \int\_0^{\text{xe}} \left( \frac{\mathbf{z}^2 \hbar}{240 \mathbf{z}\_\circ} \right) d\mathbf{z}.$$

No mobiles electric charges appear on the face of return electrode which is connected to the mass and isolated of the piezoelectric bridge, and so the Coulomb force disappears!

During this displacement "going" from 0 to ze the deformation of the piezoelectric bridge, generates a potential energy WBridge accumulated in the capacity of this bridge which follows the equation:

$$W\_{Bridge} = \int\_0^{Q\_e} \frac{Q\_F}{Cpi} dQ\_F = \left[\frac{Q\_F^2}{2^\* \cdot Cpi}\right]\_0^{Q\_e} = \frac{a\_p}{2 \cdot l\_p \cdot b\_p \varepsilon\_0 \varepsilon\_{pi}}.$$

$$\left(\frac{d31 \text{ } lp}{2ap}\right)^2 F\_{CA}^2 = \frac{a\_p}{2l\_p \cdot b\_p \varepsilon\_0 \varepsilon\_{pi}} \left(\frac{d\_{31} \text{ } lp \text{ } l\_S b\_S \pi^2 \hbar c}{480.ap}\right)^2 \left(\frac{1}{z\_\epsilon}\right)^8$$

With QF the naturally creating fixed charges on this piezoelectric structure. QF <sup>¼</sup> <sup>d</sup>31FCAlP aP Eq. (3),

Qe <sup>¼</sup> ‐QF the accumulated mobile charges on both the two surfaces of the "return" electrode when coulomb's force triggered at FCO ¼ p FCA, Cpi = electrical capacity of the piezoelectric bridge, ze the position of appearance of the Coulomb force.

From ze to 0:

For this very quick "returning" from position ze to position 0, the associated energy EReturning is:

$$E\_{Returning} = \int\_{xe}^{0} (F\_{co} + F\_{ca}) dz = E\_{Coulomb} - E\_{Casimir}$$

Thus, we see that in the balance ECasimir þ EReturning, ¼ ECoulomb. So, over a complete cycle, the energy ECasimir is conservative.!

The Coulomb energy Ecoulomb is the net energy appearing during a cycle. It is not due to any electrical energy applied, but to the consequence of vacuum energy!

If we choose zr ¼ z0, the expression of Ecoulomb is,

$$E\_{coulomb} = \left(l\_S b\_S \frac{\pi^2 \hbar c}{240} \frac{d\_{31} l\_P}{a\_P}\right)^2 \left(\frac{1}{4\pi\varepsilon\_0 \varepsilon\_r}\right)^0 \int\_{\mathcal{x}} \left(\left(\frac{1}{z\_s^4} - \frac{1}{z\_0^4}\right)\right)^2 \left(\frac{1}{z\_r + z\_0 - z\_S}\right)^2 dz\_s$$

A Coulomb force then appears between these two electrodes.

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

The position ze of appearance of this force is such that FCO ¼ p FCA, so if we choose zr ¼ z0.

Then, the zs position of appearance of this force is when:

$$\left(l\_{\rm S}b\_{\rm S}\frac{\pi^2\hbar c}{240}\frac{d\_{\rm 31}l\_{\rm P}}{2^{\ast}\,a\_{\rm P}}\left(\frac{1}{z\_{\rm s}^4}-\frac{1}{z\_{\rm 0}^4}\right)\right)^2\left(\frac{1}{4\pi\varepsilon\_0\varepsilon\_r}\right)\left(\frac{1}{2z\_{\rm 0}-z\_{\rm \rm S}}\right)^2 = pl\_{\rm S}b\_{\rm S}\left(\frac{\pi^2\hbar c}{240}\right)\frac{1}{z\_{\rm s}^4}$$

This position and evolution zs are illustrated in the following Figures 62 and 63. As a result, the movement of the movable electrode shown in Figure 62.

#### Figure 62.

Positioning of 20 Casimir cells in parallel and 10 in series. Circuit 1, Circuit 2 and Switches of circuit n ° 1 and n ° 2. Total of Casimir cells delivering a periodic current during a small part of the vibration frequency of the devices = 200!. Total des cellules = 200, width = 5 mm, length = 7 mm.

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

Figure 63.

Position of the mobile casimir electrode when the coulomb force occurs ls = 500 μm; bs = 150 μm; lp = 50 μm; bp = 150 μm; ap = 10 μm.

Note that the displacement of this mobile Casimir electrode is extremely small since it goes from 2 A° for an FCO=FCA ratio = 2 to 105 A° for a ratio of 1000.

When the electrical potential on the gate of the switches is greater than their threshold voltage, then this switch commute and the accumulated energy in the piezo electric's bridge will be used for the homogenization of the mobile charges of the bridge's electrodes connected to sources or drains of the MOS switch n°1 and the Coulomb return electrode.

During the time of this homogenization appears a current peak for a short time. So, an electrical voltage at the terminals of the self in series between switch 1 and the return electrode appears also!

So, an important point of obtaining current peaks related to the homogenization of charges appears.

His expression is: IIN ¼ � Qmn<sup>2</sup> RmCs Exp � <sup>t</sup> RmCs � � �.

With defining tc <sup>¼</sup> Rm <sup>∗</sup>Cs <sup>∗</sup> ln ð Þ<sup>2</sup> the duration with which the mobile charges equalize.

between the electrode of face 1 and the return electrode and Rm = the resistance (Ω) of the devices in series (Self + switch n°1); Cs the capacity (F) of the device constructed on face n°2 of piezoelectric bridge, return electrode).

These cyclic current peaks induce at the terminals of the coil voltage peaks whose expression is:

$$\mathcal{U}\_{\rm IN} = L\_{\rm IN} \frac{d(I\_{\rm IN})}{dt} = L\_{\rm IN} \frac{\mathcal{Q}\_{\rm mm2}}{R\_m \mathcal{C}\_s} \left( \text{Exp} \left( -\frac{t}{R\_m \mathcal{C}\_s} \right) = L\_{\rm IN} \frac{\mathcal{Q}\_{\rm mm2}}{R\_m \mathcal{C}\_s} \frac{\ln \left( 2 \right)}{t\_\varepsilon} \left( \text{Exp} \left( -\frac{t \ln \left( 2 \right)}{t\_\varepsilon} \right) \right)$$

With LIN the value of the inductance, Qmn2 the charges on the return electrode and the face n°1 of the bridge.

The only energy which is effectively used outside, during one cycle, is associated with these current and voltage peaks becomes:

$$W\_{\text{electric}} = A \text{bs} \left( \prod\_{0}^{t\_{\text{fIN}}} \frac{\text{UIN}}{\text{d}} d(t) = \frac{L\_{\text{IN}}}{2} \left( \frac{d\_{31} \text{F}\_{\text{CA}} l\_{\text{P}}}{a\_{\text{P}}} \right)^{2} \left( \frac{\ln(2)}{t e} \right)^{2} (1 - \text{Exp}(-2 \ln(2))) \right) \tag{20}$$

For example, we obtain, for an interface between Casimir's electrodes z0 ¼ 200 A°, dimensions of the Casimir electrodes (length = 500μm, width = 15μm, thickness = 10μm), dimensions of the piezoelectric bridge in PMN-PT (length = 50μm, width = 15μm, thickness = 10μm), a proportionality factor p ¼ FCO=FCA = 1000, an inductance LIN <sup>¼</sup> <sup>1</sup>:10�6H:


We remark that this energy Welectric is equal to the energy Wbridge accumulated in the bridge and is not brought by an external electrical source but is caused by the omnipresent and perpetual force of Casimir, itself controlled by a Coulomb force of opposite direction.

This Energy is less than that developed by Casimir's strength.


This energy Welectric is equal to the energyWbridge accumulated in the bridge and not brought by an external electrical source, but is caused by the omnipresent, timeless and perpetual force of Casimir, itself controlled by a Coulomb force of opposite direction.

Remember that Energy is defined as the "physical quantity that is conserved during any transformation of an isolated system." However, the system constituted by simply the MEMS device is not an isolated system.

On the other hand, the system constituted by the MEMS device plus the vacuum becomes an isolated system.

A system of 200 structures (Figure 64) gives a usable energy by second and so 750,000 peaks, Welectric ffi 60 10�<sup>3</sup> (Joule) for a coefficient of proportionality p <sup>¼</sup> FCO=FCA <sup>¼</sup> <sup>10</sup><sup>6</sup> and all switch transistors (Width = Length = 100μm, SiO2 grid thickness = 250 A°), thresholds voltage = 3 V.

This energy is not brought by an external electrical source but is caused by the omnipresent and perpetual force of Casimir, itself controlled by a Coulomb force of opposite direction.

This Coulomb force appears by the automatic switching of MOS transistors when its intensity is greater than a predetermined value and opposite to that of the Casimir force. Perspective Chapter: Device, Electronic,Technology for a M.E.M.S. Which Allow… DOI: http://dx.doi.org/10.5772/intechopen.105197

#### Figure 64.

Displacement of the mobile casimir electrode during the appearance of the coulomb force ls = 500 μm, bs = 150 μm, lp = 50 μm, bp = 150 μm, ap = 10 μm.

This technologically programmable switching of the MOS switches induces the spontaneous appearance of current peaks during a few nanoseconds, themselves inducing voltage peaks at the terminals of an inductor. When the system returns to its starting position, the Coulomb force disappears, leaving the Casimir force to deform the structure again.

The system then spontaneously enters into vibrations.

The energy balance of a cycle therefore seems to satisfy Emmy NOTHER's theorem!

### 9. Conclusions


According to this study, it would seem we can extract energy from the vacuum by the use of the Casimir force thwarted at the appropriate time by a temporary Coulomb force which makes the system return to its initial position and makes enter into vibrations the structure!!

I am fully aware that this concept may sound like incredible, but it does not seem to contradict Emmy Noether's theorem and, mathematical calculations and simulations give encouraging results and merit further study of this concept by a thesis. I am looking for a microelectronics laboratory with sufficient technological and design resources to confirm or deny this idea of a retired dreamer. The dreamer would be happy to participate in this dream of a new source of energy.
