Experimental Evidence of High Renewable Energy Employing a Symmetric Circuit with a Divergent Current Density

1. Introduction

The depletion of fossil fuels such as crude oil and natural gas has created societal energy problems. Moreover, the limited availability of high-cost crude oil has caused economic problems. To mitigate these concerns, countries are attempting to introduce renewable energies [1,2,3,4,5,6], but such alternatives cannot (in most cases) satisfy 100% of the demand. Moreover, most renewable energies fluctuate with natural climate variations, complicating the system, planning, and control methods [7]. The most promising of the existing renewable energies are solar and wind power [8,9]. Furthermore, fossil fuel consumption is a cause of rising CO₂ concentration [10,11]. The spread of renewable energy sources must be considered from both economic and technological perspectives. Highly efficient, low-cost energy generation is of major importance [12,13].

To reduce the atmospheric CO₂ concentration, existing renewable energy sources such as solar and wind power must be widely adopted. However, the integration of renewable energy into the existing power systems risks frequency and voltage disruptions. Moreover, effective performance from the existing energy sources requires many expensive storage batteries.

Japan has declared a goal of net-zero carbon by 2050, but simulations [14,15] have shown that meeting this goal might be impossible without breakthrough technologies.

The main problems with the existing renewable energy and nuclear power generation are summarized below.

1)

Renewable energy sources depend on the weather, necessitating large and expensive storage batteries. Moreover, introducing renewable energy sources to the existing energy systems will destabilize the outputs.

2)

Simulations assessing the possibility of achieving net-zero carbon in Japan have included nuclear power generation [16] without considering its dangers. For instance, the Fukushima nuclear accident in Japan in 2011 caused extensive damage.

3)

Although the seriousness of fossil fuel use differs in different countries (e.g., Japan and the United Sates [17]), introducing renewable energy incurs high costs in all countries [16].

This paper proposes a new system for stable energy production meeting the demands of Japan, i.e., the required amount of energy in the right place at the right time.

Our previously proposed energy-generating system [18,19] could not generate sufficiently large amounts of energy. The present paper proposes a new concept for generating renewable energy using a highly symmetric circuit. When certain conditions are satisfied, the circuit provides an apparent divergent current with no Joule heating to both paired light-emitting diodes (LEDs) and paired direct current (DC) motors, boosting the electric power. This power-generation phenomenon is verified in both theory and experiments. We discuss the strong agreement between the theory and experiments.

2. Method

Figure 1 is a schematic of the presented model in its initial state. The circuit employs two same-output voltage sources V and two identical loads which are not pure electrical loads, as they must output an energy of E_out per electron. As shown in the figure, this circuit system is highly symmetric. The voltage V is expressed in terms of the electric potential φ_e as follows:

$$V\equiv {\phi }_e-\left(-{\phi }_e\right)=2{\phi }_e.$$

(1)

Note that α, β, α^’ and β^’ in Figure 1 are the names of taps.

The principle of the model is shown in Figure 2, which (unlike Figure 1) includes a pair of stray capacitors C₀ embedded in a vacuum. At initial time t = 0, the stray condensers C₀ are shorted. (Note that a condenser is generally shorted at the initial state due to its voltage–current characteristic.) At some transition time t = t_c, which is essentially equal to the time constant τ_c–R (the product of the capacitance (C₀) and load resistance (R) of the load), there is a transient current (i) and an emergent electric potential (φ) defined as follows:

$$E_{out}=2e\phi ,$$

(2)

where e is the electron charge.

At time (t_c), the following condition must be satisfied:

$${\displaystyle \frac{1}{2}}{c}_0{\left(2\phi \right)}^2>R{i}^2{t}_c,$$

(3)

where all electrons penetrate the lattices of the substance. Note that resistance (R) is the internal ones of the loads in Figure 2. The right-hand side of this inequality indirectly defines the energy of the voltage source V. Thus, we can rewrite Eq. (3) as:

$$e\phi >{\displaystyle \frac{1}{2}}{k}_{B}T$$

(4)

where k_B and T denote the Boltzmann constant and temperature, respectively. In most situations, the temperature is the room temperature (i.e.$, T\approx 300 \left[K\right]$) but eq. (4) implies that Joule heating is ineffective.

As discussed later, the term $\left(e\phi \right)$ equals the kinetic energy (ε).

When the above condition is satisfied, the energies stored in the condensers are discharged. Under the energy-conservation law, energy interchange between the paired up-condenser and down-condenser in Figure 2 induces a constant current (I_E^’) along the C₀-load-to-C₀-load loop. Because the condensers C₀ are embedded in a vacuum and cannot be touched, a divergent current (I_E^’) is generated. Note that if the inequality is not satisfied, the condensers never discharge their current but if the inequality is satisfied, the voltage sources V momentary operate until the condensers begin discharging at time (t_c) and are dormant thereafter, thereby generating energy.

We now discuss the emergent electric potential (φ) after steady-state.

In the vicinity of tap α or β at t = t_c, we have:

$$d{\phi }_e\equiv \overrightarrow{E_e}·d\overrightarrow{s},$$

(5)

and

$$d{\phi }_{c0}\equiv \overrightarrow{E_{c0}}·d\overrightarrow{s},$$

(6)

where E_e and E_c0 denote the ohmic and condenser-associated electric fields, respectively.

Note that the vector (ds) aligns along the direction of (I_E^’).

Given that the condenser discharges the current, we have:

$$d{\phi }_{c0}=-E_{c0}ds.$$

(7)

Moreover,

$$d{\phi }_e=E_{e}ds.$$

(8)

However, in the vicinity of a tap, we have:

$$d{\phi }_e=d{\phi }_{c0,}$$

(9)

giving rise to

$$E_e=E_{c0}=0.$$

(10)

In the steady-state where (E_out) is nonzero, the electric potential becomes:

$$E_{out}\equiv 2e\phi ,$$

(11)

with

$$\left|\phi \right|>\left|{\phi }_e\right|,$$

(12)

This emergent electric potential (φ) is demonstrated in Figure 3.

Let us introduce a wave function contributing to the electron motion.

The general form of the Hamiltonian is:

$$H_{total}\equiv H_G+H_{eff}=\left(-{\displaystyle \frac{{ħ}^2}{2m}}{\displaystyle \frac{{\mathrm{d}}^2}{d{x}^2}}\pm e\phi \right)+H_{eff},$$

(13)

where the first and second terms describe the center-of-mass motion and the net electron–electron interaction, respectively. As the emergent energy (eφ) is sufficiently large, the following assumption holds:

$$H_G\gg H_{eff}.$$

(14)

This condition guarantees the one-particle picture. If Eq. (4) is satisfied, the Hamiltonian and electron motion can be approximated as one-dimensional because lattice disturbance and electron scattering phenomena are absent.

From an alternative perspective, we can assume a uniform electrical potential (φ) at tap α or β, implying that each electron has approximately the same electrical potential (φ). As (φ) cannot coexist with a Coulomb interaction potential (φ_M), the Hamiltonian (H_eff) is sufficiently small. Moreover, whether the potential (φ) is macroscopic or microscopic cannot be discerned and the absence of net heating implies no Fermi statistics, which includes a temperature, even at local positions.

The Hamiltonian at tap α is:

$$H_{\alpha }=-{\displaystyle \frac{{ħ}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+(-e\phi ),$$

(15)

where m denotes the mass of an electron.

As energy is conserved, we have:

$${\displaystyle \frac{{ħ}^2{k}_{\alpha }^2}{2m}}\equiv \left|-e\phi \right|\equiv {\epsilon }_{\alpha },$$

(16)

where k is the wavenumber.

Thus,

$$k_{\alpha }=\pm {\displaystyle \frac{\sqrt{2me\phi }}{ħ}}.$$

(17)

Here, we associate the positive and negative values of (k_α) with up-spin and down-spin, respectively. Thus, the wavefunctions are

$${\psi }_{\alpha \uparrow }=\left|\psi \right|\exp \left(j{\displaystyle \frac{\sqrt{2me\phi }}{ħ}}x\right),$$

(18)

and

$${\psi }_{\alpha \downarrow }=\left|\psi \right|\exp \left(-j{\displaystyle \frac{\sqrt{2me\phi }}{ħ}}x\right),$$

(19)

respectively. Note that (j) is the imaginary unit. Similarly to tap α, tap β satisfies the following Hamiltonian:

$$H_{\beta }=-{\displaystyle \frac{{ħ}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+(+e\phi ).$$

(20)

By energy conservation, we have:

$${\displaystyle \frac{{ħ}^2{k}_{\beta }^2}{2m}}\equiv \left|+e\phi \right|\equiv {\epsilon }_{\beta },$$

(21)

where

$$k_{\beta }=\mp {\displaystyle \frac{\sqrt{2me\phi }}{ħ}}.$$

(22)

Here, positive and negative (k_β) denote down-spin and up-spin, respectively. Note that the potential $(\pm e\phi )$ in the Hamiltonians is the Coulomb force. That is, the taps with positive and negative potentials exert an attractive force (–F) and a repulsive force (+F) on the electron, respectively. These forces are effective only when the electron motions form in negligible lattice disturbance (i.e., negligible Joule heating). Note that as these forces concern only the center-of-mass motion, inter-electron Coulomb interactions can be ignored as previously discussed.

In summary, the wave functions of α and β are dual; that is:

$${\psi }_{\beta \downarrow }=\left|\psi \right|\exp \left(j{\displaystyle \frac{\sqrt{2me\phi }}{ħ}}x\right),$$

(23)

and

$${\psi }_{\beta \uparrow }=\left|\psi \right|\exp \left(-j{\displaystyle \frac{\sqrt{2me\phi }}{ħ}}x\right).$$

(24)

The Slater determinant:

$${\psi }_E\equiv {\displaystyle \frac{1}{\sqrt{2}}}{\left|\psi \right|}^2({\psi }_{\alpha \uparrow }{\psi }_{\beta \downarrow }-{\psi }_{\alpha \downarrow }{\psi }_{\beta \uparrow }),$$

(25)

normalized such that:

$$\int {\left|\psi \right|}^{2}dv\equiv 1$$

(26)

becomes

$${\psi }_E={\displaystyle \frac{1}{\sqrt{2}}}{\left|\psi \right|}^2\{\exp \left(j{\displaystyle \frac{2\sqrt{2me\phi }}{ħ}}x\right)-\exp \left(-j{\displaystyle \frac{2\sqrt{2me\phi }}{ħ}}x\right)\}.$$

(27)

The first and second terms of the above equation correspond to the left and right sides of the circuit in Figure 3, respectively. The first and second terms are considered to be related through the Einstein–Podolosky–Rosen correlation [20].

Here, we consider only the first term of Eq. (27).

The uncertainty relation gives

$$∆\theta ∆N\approx 1$$

Because the phase (θ) is constant, the uncertainty $(∆N)$ becomes infinite, defying the classical physical motion of an electron from one tap to the other. The wavefunction of the left part of the circuit is:

$${\psi }_L\equiv \left|{\psi }_L\right|\mathrm{e}\mathrm{x}\mathrm{p}\left[j{\displaystyle \frac{2\sqrt{2me\phi }}{ħ}}x\right].$$

(28)

The probability density (j_E) of the flow:

$$j_E={\displaystyle \frac{ħ}{2mj}}({\psi }_L^{*}{\displaystyle \frac{d{\psi }_L}{dx}}-{\psi }_L{\displaystyle \frac{d{\psi }_L^{*}}{dx}})$$

(29)

is solved as:

$$j_E={\displaystyle \frac{ħ}{2m}}{\displaystyle \frac{4\sqrt{2me\phi }}{ħ}}{\left|{\psi }_L\right|}^2.$$

(30)

Herein, we separately consider paired LEDs or paired DC motors as the loads.

1)

DC motor loads

From the normalization condition:

$$\int {\left|{\psi }_L\right|}^{2}dv\equiv 1,$$

(31)

we have

$${\left|{\psi }_L\right|}^2{\lambda }^3=1,$$

(32)

where λ is the wavelength of an electron.

The wavenumber (k) is defined as:

$$k\equiv {\displaystyle \frac{1}{\lambda }}$$

(33)

Thus,

$${\left|{\psi }_L\right|}^2\approx k^3$$

(34)

The wavenumber is also derived as:

$$k={\displaystyle \frac{2\sqrt{2me\phi }}{ħ}}$$

(35)

from which the flow probability density (j_E) is obtained as:

$$j_E={\displaystyle \frac{4\sqrt{2me\phi }}{2m}}({{\displaystyle \frac{2\sqrt{2me\phi }}{ħ}})}^3$$

(36)

That is,

$$j_E={\displaystyle \frac{64m^2{e}^2}{m{ħ}^3}}{\phi }^2$$

(37)

To obtain the net current density [A/m²], we multiply Eq. (37) by the electronic charge (e):

$$j_{E=}^{\text{'}}{\displaystyle \frac{64m{e}^3}{{ħ}^3}}{\phi }^2\approx 2\times {10}^{17}{\phi }^2$$

(38)

Here, the area (S_i) is considered as a unit cell; for example,

$$S_i\cong {(1.5\times {10}^{-9})}^2 \left[{\mathrm{m}}^2\right].$$

(39)

Importantly, the net observed current ${(I}_E^{\text{'}})$ is then derived as:

$$I_E^{\text{'}}\approx 0.45{\phi }^2 \left[\mathrm{A}\right].$$

(40)

2)

LED loads

Under the normalization condition:

$$\int {\left|{\psi }_L\right|}^{2}dv\equiv 1,$$

(41)

we again have:

$${\left|{\psi }_L\right|}^2{\lambda }^3=1,$$

(42)

but here we consider the momentum of a photon with wavenumber [21]:

$$k={\displaystyle \frac{2\pi }{\lambda }}$$

(43)

The photon imparts its momentum to an electron or hole; that is:

$${\left|{\psi }_L\right|}^2={\displaystyle \frac{1}{8{\pi }^3}}{k}^3$$

(44)

and

$$j_E={\displaystyle \frac{4\sqrt{2me\phi }}{2m}}({{\displaystyle \frac{2\sqrt{2me\phi }}{ħ}})}^3{\displaystyle \frac{1}{8{\pi }^3}}={\displaystyle \frac{64m^2{e}^2}{m{ħ}^3}}{\displaystyle \frac{1}{8{\pi }^3}}{\phi }^2$$

(45)

Similarly to the DC motor, the LED loads obtain a net current density of:

$$j_E^{\text{'}}={\displaystyle \frac{64m{e}^3}{{ħ}^3}}{\displaystyle \frac{1}{8{\pi }^3}}{\phi }^2\approx 8.0\times {10}^{14}{\phi }^2 [\mathrm{A}/{\mathrm{m}}^2].$$

(46)

As the net current is contributed by an electron and a hole, we have:

$$j_{E,net}^{\text{'}}\approx 2\times j_E^{\text{'}}=1.6\times {10}^{15}{\phi }^2 [\mathrm{A}/{\mathrm{m}}^2].$$

(47)

Within the area (S_i) of a unit cell, e.g.,

$$S_i\cong {(5.0\times {10}^{-9})}^2 \left[{\mathrm{m}}^2\right].$$

(48)

the net current is

$$I_E^{\text{'}}\approx 0.04{\phi }^2\left[\mathrm{A}\right].$$

(49)

Next, the DC-motor loads must satisfy two conditions. Let us consider these conditions:

The general motion equation of a motor is [22] as follows:

$$V_a=R_a{I}_a+L{\displaystyle \frac{d{I}_a}{dt}}+E,$$

(50)

where R_a [Ω], I_a [A], L [H], and E [V] denote the inner resistance, working current, inductance, and reverse electromotive force, respectively.

Let us first obtain the general solution from Eq. (50). To obtain this solution, setting the nonhomogeneous term ${(V}_a-E)$to zero, the general solution is immediately obtained as:

$$I_a=I_a\left(0\right)\exp (-{\displaystyle \frac{R_a}{L}}t).$$

(51)

At t = 0 when all currents flow into the condensers C₀, the initial current in the motor is zero; that is:

$$I_a\left(0\right)\equiv 0.$$

(52)

Moreover, the special solution of Eq. (50) is derived by:

$$I_a\equiv A=const.$$

(53)

Thus, by the substitution to eq. (50),

$$A={\displaystyle \frac{V_a-E}{R_a}},$$

(54)

and thus the net solution, i.e., the sum of the general solution and the special solution, is:

$$I_a=I_a\left(0\right)\exp \left(-{\displaystyle \frac{R_a}{L}}t\right)+{\displaystyle \frac{V_a-E}{R_a}}=0+{\displaystyle \frac{V_a-E}{R_a}}.$$

(55)

The transient equation of the motors in our system is then given by:

$$V_a=R_a{I}_a+E,$$

(56)

where

$$E\equiv k_E{\Omega }_m.$$

(57)

In this equation, the proportionality constant (k_E) depends on the magnetic flux, and (Ω_m) is the rotational velocity.

The current (I_a) is considered as the constant current under zero load (I_a0).

Under the working condition:

$$E_{out}>k_{B}T$$

(58)

of our system, the motor must satisfy:

$$R_a{I}_{a0}<k_E{\Omega }_{m0},$$

(59)

where Ω_m0 is the speed under zero load.

We now derive the second mandatory condition of the motors. To this end, we discuss the net voltage of the motor. At t = t_c when the voltages between the condenser and voltage source are canceled, V_a = 0. From Eq. (56), we then have:

$$E\equiv E_c=-R_a{I}_{a0},$$

(60)

meaning that (I_a0) is an induced loop current not induced by the voltage source.

At $t\ge t_c$ when the current is discharged by the condensers C₀,

$$V_a\equiv V_c=E<0,$$

(61)

where V_c is the discharging voltage of the condenser.

As discussed below, the above equation neglects the Joule heating term.

The motor voltage becomes the superposition of (E_c) and (E), as shown in Figure 4. Thus, the net energy of an electron is:

$$E_{eff}\equiv e\left|E+E_c\right|\equiv 2e\phi .$$

(62)

Based on the abovementioned discussion, the motors must satisfy:

$${\displaystyle \frac{1}{2}}L{I}_{a0}^2>R_a^{\text{'}}{I}_{a0}^2{\tau }_{LR},$$

(63)

where $R_a^{\text{'}}$ and ${\tau }_{LR}$ denote the impedance input to the other motor during the current discharge, which may include the internal resistance of the voltage source V, and ${(\tau }_{LR})$ is the time constant of $\left(L\right) \mathrm{a}\mathrm{n}\mathrm{d} \left(R_a^{\text{'}}\right)$. The left side of Eq. (63) is the coil discharge of the motor, but because the motors are paired in our system, they reciprocally provide and receive energy. Provided that the second condition above is satisfied, the Joule heating term is absent and the motor receives a relatively large input voltage, as discussed in the Results section. Moreover, we will show that because the voltage of a motor is:

$$E_{eff}\equiv e\left|E+E_c\right|\equiv 2e\phi >V=2{\phi }_e,$$

(64)

implying that the voltage sources V are nonoperational, the paired motors repetitively store and discharge energy.

The macroscopic current (I_a0) can coexist with the microscopic current ${(I}_E^{\text{'}})$ in the load but the two currents are not superposed, so an additive current does not appear. Thus, if a current meter is connected in series with a motor, it records either the current (I_a0) or the current $\left(I_E^{\text{'}}\right)$ at each moment, presenting widely fluctuating currents to the experimental observer. Accordingly, we set a detour for the current $\left(I_E^{\text{'}}\right)$ in the system with paired DC motors (Figure 5). In the Results section, we will discuss the importance of:

$$I_E^{\text{'}}\gg I_{a0}.$$

(65)

Here, let us introduce the output electric power (W). The incremental value of (W) is

$$dW\equiv I_E^{\text{'}}\left(0\right)\times 2d\phi ,$$

(66)

where

$$I_E^{\text{'}}\left(0\right)\equiv 0.45{\left({\displaystyle \frac{E+E_c}{2}}\right)}^2=const.$$

(67)

Given the voltage–current characteristics Eqs. (40) and (49), the derivative derived from these characteristics inevitably provides an electric resistance (R₀). Thus, we have

$$2d\phi =R_{0}d{I}_E^{\text{'}}.$$

(68)

Thus,

$$dW=I_E^{\text{'}}\left(0\right)R_{0}d{I}_E^{\text{'}}.$$

(69)

Given $I_E^{\mathrm{\text{'}}}=0.45{\phi }^2$, we obtain

$$dW=I_E^{\text{'}}\left(0\right)R_0\times d\left(0.45{\phi }^2\right)=0.45I_E^{\text{'}}\left(0\right)R_0\times 2\phi d\phi .$$

(70)

Integrating both sides of Eq. (70) gives

$$W=0.45R_0{I}_E^{\text{'}}\left(0\right){\phi }^2 \left[\mathrm{W}\right].$$

(71)

Provided that the conditions in our system are satisfied, Eq. (71) is not a Joule-heating expression. The absence of Joule heating can be explained by the dead voltage sources, as frequently described in this paper. The voltage sources are only temporarily active in the vicinity of the initial time.

3. Results

We first present the results of the paired-LED system with:

$$E_{out}\equiv 2e\phi$$

(72)

or

$$E_{out}\equiv ħ\omega ,$$

(73)

where ω is the angular frequency. The divergent current is

$$I_E^{\text{'}}=0.04({{\displaystyle \frac{ħ\omega }{2e}})}^2.$$

(74)

Given the approximate wavelength (λ) of a blue LED:

$$\lambda =460 \left[\mathrm{nm}\right].$$

(75)

the angular frequency is calculated as:

$$\omega ={\displaystyle \frac{2\pi c}{\lambda }}\approx 4.1\times {10}^{15} [\mathrm{rad}/\mathrm{s}].$$

(76)

Thus,

$$I_E^{\text{'}}=0.04{\phi }^2=0.04({{\displaystyle \frac{ħ\omega }{2e}})}^2\approx 0.04\times ({1.34)}^2\approx 71 [\mathrm{mA}].$$

(78)

The voltage is

$$2\times 1.34=2.68 \left[\mathrm{V}\right].$$

(79)

The above current and voltage are valid.

The results of the LED system are summarized in the following three figures: a schematic configuration of the circuit, a photograph of the initial experimental setup, and a photograph of the experimental result.

Figure 6-1. Schematic of divergent current $I_E^{\mathrm{\text{'}}}$ generation in the system with LED loads. As described in the main text, the divergent current $I_E^{\mathrm{\text{'}}}$ drives the LEDs and the voltage sources die at steady-state.

Figure 6-1. Schematic of divergent current $I_E^{\mathrm{\text{'}}}$ generation in the system with LED loads. As described in the main text, the divergent current $I_E^{\mathrm{\text{'}}}$ drives the LEDs and the voltage sources die at steady-state.

Figure 6-2. Initial setting of the experiment for confirming operation of the LED pair. The setup includes two power supplies (voltage sources) and two blue LEDs.

Figure 6-2. Initial setting of the experiment for confirming operation of the LED pair. The setup includes two power supplies (voltage sources) and two blue LEDs.

Figure 6-3. Experimental result of the system with two LED loads. The paired-LED is working and the current in the voltage sources is almost zero, confirming the appearance of the divergent current $I_E^{\mathrm{\text{'}}}$ and decay of the net voltage sources.

Figure 6-3. Experimental result of the system with two LED loads. The paired-LED is working and the current in the voltage sources is almost zero, confirming the appearance of the divergent current $I_E^{\mathrm{\text{'}}}$ and decay of the net voltage sources.

We now present the results of the paired motor system.

Table 1 lists the specifications of the employed DC motor, and Figs. 7 and 8 show a potable tester and the employed paired motors, respectively.

Figure 7. Photograph of our tester, included for the reasons given in the Discussion.

Figure 7. Photograph of our tester, included for the reasons given in the Discussion.

Figure 8. Photograph of the paired DC motors (MABUCHI motor RE-280RA), marked with black dots to confirm whether the motors rotate during the experiment.

Figure 8. Photograph of the paired DC motors (MABUCHI motor RE-280RA), marked with black dots to confirm whether the motors rotate during the experiment.

Figure 9 and Figure 10 present the three stages of the paired motor experiments: the circuit configurations, the preparations of the experiments, and the experimental results.

Figure 9-1. Schematic of the circuit for measuring an emergent electric potential. The voltage of the voltage source is compared with that of the DC motor, which implies the emergent electric potential φ.

Figure 9-1. Schematic of the circuit for measuring an emergent electric potential. The voltage of the voltage source is compared with that of the DC motor, which implies the emergent electric potential φ.

Figure 9-2. Experimental setup for measuring the emergent electric potential. The voltage sources are two stabilized power supplies, one with a red light display; the other with a blue display. The two DC motors are not rotating at this time, as evidenced by the stationary black dots. The emergent electric potential is measured by a voltmeter (foreground).

Figure 9-2. Experimental setup for measuring the emergent electric potential. The voltage sources are two stabilized power supplies, one with a red light display; the other with a blue display. The two DC motors are not rotating at this time, as evidenced by the stationary black dots. The emergent electric potential is measured by a voltmeter (foreground).

Figure 9-3. A result of the experimental setup in Figure 9-2. When the voltage source displays 3.0 V, the voltage associated with the emergent electric potential is 5.17 V, approximately 40% larger than the source voltage.

Figure 9-3. A result of the experimental setup in Figure 9-2. When the voltage source displays 3.0 V, the voltage associated with the emergent electric potential is 5.17 V, approximately 40% larger than the source voltage.

Figure 10-1. Schematic of the experiment for harvesting the current $I_E^{\mathrm{\text{'}}}.$As in Figure 5, a detour avoids interference between the induced macroscopic current $I_{a0}$ and the microscopic current $I_E^{\mathrm{\text{'}}}$. As the mathematical form of the current $I_E^{\mathrm{\text{'}}}$ is independent of the load $R_0$ and differs from Ohm’s law, the current flows even when the load $R_0$ is large. For the same reason, the load resistance $R_0$ generates no Joule heating.

Figure 10-1. Schematic of the experiment for harvesting the current $I_E^{\mathrm{\text{'}}}.$As in Figure 5, a detour avoids interference between the induced macroscopic current $I_{a0}$ and the microscopic current $I_E^{\mathrm{\text{'}}}$. As the mathematical form of the current $I_E^{\mathrm{\text{'}}}$ is independent of the load $R_0$ and differs from Ohm’s law, the current flows even when the load $R_0$ is large. For the same reason, the load resistance $R_0$ generates no Joule heating.

Figure 10-2. Initial setup of the experiment for harvesting the current $I_E^{\mathrm{\text{'}}}$. Shown are the two voltage sources, the paired motors, a current meter (range 10 A), and a 1.0-MΩ resistor connected by the red and yellow leads (in the vicinity of the left motor).

Figure 10-2. Initial setup of the experiment for harvesting the current $I_E^{\mathrm{\text{'}}}$. Shown are the two voltage sources, the paired motors, a current meter (range 10 A), and a 1.0-MΩ resistor connected by the red and yellow leads (in the vicinity of the left motor).

Figure 10-3. Experimental result when the initial input voltage is ~6.0 V. The divergent current $(I_E^{\mathrm{\text{'}}}$ =5.78 A) is much larger than the induced current$(I_{a0}\approx 0.24$A).

Figure 10-3. Experimental result when the initial input voltage is ~6.0 V. The divergent current $(I_E^{\mathrm{\text{'}}}$ =5.78 A) is much larger than the induced current$(I_{a0}\approx 0.24$A).

Figure 10-4. Experimental result when the initial input voltage is 10.0 V. The current meter reads $I_E^{\mathrm{\text{'}}}$>14 A, much larger than the current $I_{a0}$. Even when the load is very large (1.0 MΩ), a substantially large current is flowing. Note that no Joule heating occurs in the motors, and a relatively large input voltage is allowed. That is, these motors can be repeatedly used even when the input voltage exceeds 3.0 V. .

Figure 10-4. Experimental result when the initial input voltage is 10.0 V. The current meter reads $I_E^{\mathrm{\text{'}}}$>14 A, much larger than the current $I_{a0}$. Even when the load is very large (1.0 MΩ), a substantially large current is flowing. Note that no Joule heating occurs in the motors, and a relatively large input voltage is allowed. That is, these motors can be repeatedly used even when the input voltage exceeds 3.0 V. .

Figure 10-5. Experimental result when the initial input voltage is 12.0 V. The divergent current $I_E^{\mathrm{\text{'}}}$is out-of-range, implying that it exceeds 20.0 A through the 1.0-MΩ load.

Figure 10-5. Experimental result when the initial input voltage is 12.0 V. The divergent current $I_E^{\mathrm{\text{'}}}$is out-of-range, implying that it exceeds 20.0 A through the 1.0-MΩ load.

Figure 11. Circuit configuration after inserting another motor as a detour. Note that the polarity (i.e. the direction) of each motor is important. That is, the polarity of M’ must be reversed from that of the paired motors and motor M’ rotates oppositely to the paired motors. If configured otherwise, M’ will not rotate. The photograph is omitted because motor M’ is not easily distinguishable; therefore, the photograph is noninformative.

Figure 11. Circuit configuration after inserting another motor as a detour. Note that the polarity (i.e. the direction) of each motor is important. That is, the polarity of M’ must be reversed from that of the paired motors and motor M’ rotates oppositely to the paired motors. If configured otherwise, M’ will not rotate. The photograph is omitted because motor M’ is not easily distinguishable; therefore, the photograph is noninformative.

Table 2 compares the experimentally observed currents $I_E^{\text{'}}\left(0\right)$ with the theoretical currents determined as:

$$I_E^{\text{'}}\left(0\right)\equiv 0.45{\left({\displaystyle \frac{E+E_c}{2}}\right)}^2$$

(80)

where (E + E_c)/2 is the emergent electric potential (φ) with

$$\phi >{\phi }_e=V/2$$

(81)

As the emergent electric potential must not be significantly larger than the input electric potential (φ_e), we approximate

$${\displaystyle \frac{E+E_c}{2}}\approx 1.2{\phi }_e$$

(82)

as a compromise. As shown, Table 2 confirms sufficient agreements between the experimental and theoretical values.

Table 3 lists the generated electric power (W) for different input voltages (2φ_e)_. Note that, in this table, (φ) in eq. (71) is again defined by eq. (82) but acutally it is possible to take a larger value. Thus, the values in Table 3 might be larger accutally. We should consider the values of Table 3 as the minimum values. This will be described in Discussion section. The precise estimation of (φ) in this case will be a follow-up. However, the values of Table 3 are still sufficiently large, which is almost equal to that of a nuclear power station. This implies that the values of Table 3 have already provided significant technical merits.

For comfirmation, we validate Eq. (71); that is:

$$W=0.45R_0{I}_E^{\text{'}}\left(0\right){\phi }^2$$

(83)

in the paired motor system. The electric power of the LED is obtained simply by replacing coefficient 0.45 [A/V²] with 0.04 [A/V²] as:

$$W_{LED}=0.04R_0{I}_E^{\text{'}}\left(0\right){\phi }^2$$

(84)

Assuming

$$I_E^{\text{'}}\left(0\right)=71 \left[\mathrm{mA}\right]$$

(85)

and

$$\phi =1.34 \left[\mathrm{V}\right],$$

we obtain      (86)

$$R_0={\displaystyle \frac{2\times 1.34V}{71mA}}\approx 38 [\Omega ].$$

(87)

from which $W_{LED}$ is calculated as:

$$W_{LED}\approx 0.193$$

(88)

On the other hand, the direct electric power W’_LED is defined as simply

$$W_{LED}^{\text{'}}\equiv I_E^{\text{'}}\times 2\phi$$

(89)

Then we get:

$$W_{LED}^{\text{'}}\approx 0.190 \left[\mathrm{W}\right].$$

(90)

Thus

$$W_{LED}\approx W_{LED}^{\text{'}}$$

(91)

validating the expression of W_LED. Considering that eq. (83) (i.e., eq. (71)) differs only in the coefficient (i.e., $0.04\to 0.45$), it is allowed to mention that the above result (91) also validates eq. (71).

Figure 11 shows the circuit configuration when another motor is employed as the detour. The detour motor operated only when its polarity was reversed from that of the paired motors. In this configuration, the additional motor exchanged energy with the left member of the motor pair. The exchanged energy is:

$$U_M={\displaystyle \frac{1}{2}}L{I}_E^{\text{'}2},$$

(92)

where E [V] are exchanged among $\left(E+E_c\right)$ [V].

4. Discussion

4.1. Why Can a Large Current Flow into the 1.0-MΩ Resistor?

A circuit containing a resistance (R) typically exhibits ohmic characteristics, meaning that its current depends on (R). However, in the present study,

$$I_E^{\text{'}}\left(0\right)\equiv 0.45{\left({\displaystyle \frac{E+E_c}{2}}\right)}^2=const.,$$

(80)

which is independent of the resistance (R), meaning that the current flows through any resistor. The large current flow through the 1.0-MΩ resistor is explained not only by this phenomenon, but also by the absence of Joule heating under the imposed conditional inequalities.

4.2. Condition and Selection of the Paired Motor

To ensure that the paired DC motors satisfy the conditional inequalities (59) and (63), we recommend motors with relatively large inductances. However, inductors with excessive inductance should be avoided because they increase the internal resistance. Therefore, the paired DC motors must be carefully selected. This paper employs the previously discussed MABUCHI motor RE-280RA [23]. A trial experiment using coreless motors as the paired DC motors failed to display the phenomenon described here, possibly because coreless motors with relatively small inductances cannot satisfy the requisite conditions (59) or (63).

4.3. Why Is a Relatively Large Resistance Needed when Harvesting the Current?

To answer this question, we revisit Eq. (80):

$$I_E^{\text{'}}\left(0\right)\equiv 0.45{\left({\displaystyle \frac{E+E_c}{2}}\right)}^2.$$

which implies that the voltage between taps α and β is $(E+E_c)$ [V]. Thus, when harvesting the current through a detour, the resistance (R₀) should be sufficiently large to satisfy (see Figure 10-1)

$$\left|{\phi }_{\alpha ,\beta }\right|\ge \left|{\displaystyle \frac{E+E_c}{2}}\right|$$

(93)

The left-hand-side of Eq. (93) is the absolute value of a further emergent electric potential of taps α or β when current $I_E^{\text{'}}\left(0\right)$ flows through the detour. That is, the current $I_E^{\text{'}}\left(0\right)$ remains constant but a voltage of the resistance (R₀) along the detour varies. When the condition (93) is satisfied, the current $I_E^{\text{'}}\left(0\right)$ chooses the path along the detour instead of the path of the motor.

When a current meter is placed along the detour, the electric potential at taps α and β should also satisfy the above condition. According to Figure 10-1, a relatively large load (R₀) will enlarge the electric potential (φ_α). Moreover, our tester with a relatively large flow divider of a 10-A range is justified for boosting the electric potential (φ_β). (Note that a digital tester generally deals with data as the voltages (Not the currents)).

Moreover, the phenomenon detected in the many experiments of this paper could not be detected by an autorange detector. Such a detector should be employed in a follow-up study, but an internal resistance that varies with the electrical signals might violate the above condition. Perhaps, there is a macroscopic current (i_A) in autotange meter and thus this current (i_A) might interefere with the microscopic current ${(I}_E^{\text{'}})$. Therefore, an autorange meter was not employed in our present experiments (Figure 7).

When another motor M’ is employed as a detour, energy exchanges between the inductors give rise to a constant current $\left(I_E^{\text{'}}\right)$. As the additional motor has a very small internal resistance, it demonstrates the potential of supplying the current $\left(I_E^{\text{'}}\right)$ to a small resistance via the observed phenomenon.

4.4. Recap of the Key Concept

This paper describes a novel concept and a highly symmetric circuit for renewable energy generation. A circuit satisfying the requisite conditions generates a divergent current with no Joule heating. The divergent current drives both paired-LED and paired-DC-motor systems, generating large amounts of energy. The phenomenon is demonstrated both theoretically and experimentally, with good agreement between the theoretical and experimental results.

4.4. Significances of This Paper

Many countries are attempting to mitigate the pending energy crisis with solar cells, wind power, and biomass energy sources. However, natural energy produced by renewable energy sources inevitably fluctuate and destabilize the energy system. Stability can be restored only by introducing many storage batteries that increase the costs. More seriously, fossil fuel energies are major CO₂ emitters, whereas nuclear power stations (which release no CO₂) destroy the environment with radioactive materials and their wastes.

Our proposed methodology generates a large amount of energy using a highly symmetric circuit. The main contributions are listed below.

1)

The net generated energy almost equals that of a nuclear power station, implying a sufficiently high energy density.

2)

The system can be introduced at low cost (<10,000 yen).

3)

The system generates no dangerous substances.

4)

This system can provide the required amount of energy at the right time at the right place.

In summary, our simple-to-build system generates comparable energy to a nuclear power station.

5. Conclusion

This study proposes and verifies (theoretically and experimentally) a new circuit-based renewable-energy source. The system outputs a very high electric power and high energy density without natural fluctuations or dangerous substance production. That is, it harvests the required amount of energy at the proper time at the proper place.

Various motor types were experimentally trialed in the paired-motor system, but a more systematic experiment would consolidate the aforementioned conditions. Moreover, the capacitance of the stray capacitor is expected to slightly vary with humidity and atmospheric pressure. This expectation must be clarified in future work. Finally, the inapplicability of the autorange meter must be rationalized.