A*(2)=A x A
examples; 2*(2)=4, 4*(1/2)=2, (4/9)*(1/2)=2/3, 3(4/9)*(1/2)=2, k4*(1/2)=2k

//About the correlativity of energy multiplying phenominon of Tesla Coil and Hutchison effect

Generally, we make the tank circuit to oscillate cosine-curved electric vibration. In use of Tesla Coil, it isn't an exception.
//But, every tank circuit can't escape from the impediment of consuming energy that resitance causes. Usually, we tend to eliminate resistance from tank circuit to reduse the loss of heat.
//I solved the differential equation which describes the action of series LCR circuit(a part of Tesla Coil). The answer showed that tank circuit shows strange behavior when resistance of tank circuit was larger than a certain value. Under this condition, it is expected that Tesla Coil will multiply energy, or absorb deficient energy in the capacitive space of capacitor.
//I think this is the cause of Hutchison effect, and be sure of reapperance of this effect to regulate the resistance of general Tesla Coil(with spark gap).

// I . Behavior of reconstructed TC

//fig. 1 Reconstructed Tesla Coil

//This is the plan of TC's new reguration. Resistance R is larger than 2 (L/C)*(1/2) , and the secondary inductor is connected to the other resistor R' not the metallic globe or ground. Under this condition, all supplied energy from the battery V0 will be changed into thermal energy at each resistors.

/ 1. " Charge C "
//First, the switch is conected to the point of contact (1), and V0 charges the capacitor C . Electrical current " I " carries electric charge from V0 to C, but the electric energy is also spent at the resistor R1.
// The capacitive voltage of C ( VC ) and the electric current " I " can be expressed by V0, R1, C as following .

//VC= -V0+V0e*(-t/CR1)

//I =V0e*(-t/CR1) / R1

//The sum of accumulated electrostatic energy EC(suppose infinite time) is equal to -CV0*(2) / 2.
//And the sum of electric power ER1(t ; 0~infinite) which is consumed at the resistor R1 becomes following. Electric power WR1(= -R1I*(2) ) is the differential of ER1.

//ER1 was equal to EC. As a result, the total energy Esply which V0 supplies becomes twice as much as absolute quantity of EC.

Esply= -EC -ER1=CV0*(2) ........[1]

//Esply is CV0*(2) .

//2. " Discharge C "

//Next, after full-charging of C, turn the connecting point of the switch from (1) to (2). Capacitor C will discharge its electrostatic energy, and another electric current I1 will flow into inductor L and resistor R. Action of this part(tank circuit) is expressed as following differential equations. In these equations, VL is the inducted voltage of L, and VR is the resistive voltage of R.

//VC+VL+VR=VC - LI1' - RI1=0

//I1= - CVC'

//Answers are following ( using "w" instead of Greece "omega", "w" is angular frequency).

VC= -V0e*(-wt)

//VL= -CLV0w*(2)e*(-wt)


//I1= -wCV0e*(-wt)

//w=(R/2L) +or- (R*(2)/4L*(2) - 1/CL)*(1/2)
//("w" is the answer of equation "CLw*(2) - CRw +1 = 0 ")

//From these answers, each electric power(WC, WL, WR) and consumed(or discharged) energy(EC, EL, ER, ER') become following. But, providing that EL must be consumed at R', and EL must be equal to -ER'.



//WR= -R(wCV0)*(2)e*(-2wt)


//EL= L(wCV0)*(2)/2 (= -ER')

//ER= -Rw(CV0)*(2)/2

//Therefore, all generated thermal energy Egnrt becomes "|ER'|+|ER|+|ER1|(=|EL|+|ER|+|ER1|)".

//Egnrt=|ER'|+|ER|+|ER1|=L(wCV0)*(2)/2 + Rw(CV0)*(2)/2 + CV0*(2)/2

////////=C(CLw*(2) + CRw + 1)V0*(2)/2


//Then, from the results of [1] and [2] , evaluate "Egnrt - Esply". Define it as "E".

sply=Rw(CV0)*(2) - CV0*(2)=(CRw - 1)CV0*(2)


//According to this result, if angular frequency "w" were actual number at least, E becomes larger than zero. When R was equal to or larger than 2 (L/C)*(1/2), this condition will be satisfied. It means that this circuit multiplies enrgy. E is the profit.
//Especially, when R=2 (L/C)*(1/2), w becomes (1/CL)*(1/2). Under this condition, E becomes CV0*(2), it is as much as supplied enrgy Esply.

//I named this equipment, "The Twisty Energy Multiplier". Because this circuit works like changing debt into profit. Essencially, EL is the energy of debt, but secondary inductor and R' changes its attribution from "debt" into "profit".
//It sounds like "falsifying of account" for me. I think it is like a certain countorie's national bank buying its national debt. And also, I feel it is "twisty" to change attribution of EL from "debt" into "profit", or rewriting its mark from "minus(-)" into "plus(+)". So, I gave it the name "twisty".

//3. A plan to realize continuous motion

fig 2. "The Twisty Energy Multiplier"

Condition ;/R=2 (L/C)*(1/2)

/////Under this condition,


/////VC=VL= -V0e*(-wt), VR= -2VL= -2VC
/////(shown as the size and directions of each arrows at fig.1)

/////I1= -2CV0e*(-wt)/R

//|EC|=|EL|=|ER'|=|ER1|=CV0*(2)/2, |ER|= CV0*(2)




Generated heat can be changed into electricity by using steam engine or Peltier effect(or thermocouple).
//This circuit(fig2) can "conduct" continuously for the help of spark gap which is used as an excellent substitute for the switch of fig.1.
/When C was charged enough by the battery V0, it opens spark gap and discharges its electrostatic enrgy for L and R. And when spark gap was closed for the lack of electrostatic energy of C, battery V0 begins to charge C again. This circuit works continuously with the repetition of alternative these two actions.
//Frankly to say, this is nothing but a Tesla Coil. Just I found its potential ability.

//II. Correlativity with Hutchison effect

fig.3 A plan for the purpose of reappearing of Hutchison effect

//This is the "suspected" plan which may be able to reappear Hutchison effect in the capacitive space of capacitor C'. On this plan, L' is a secondary inductor which is connected to the primary inductor L indirectly, and, a capacitor C' and a resistor R'(basically, this is a large resistance) is connected to L' in series.

Conditions about this circuit are folowing(condition 1).

//R';voluntary (depends on proper charging speed of C)

//R=2 (L/C)*(1/2), R'=2 (L'/C')*(1/2)


//Provided that L' receives magnetic energy of L in its entirety.

//According to this provision, energy of L' (EL') must be equal to -EL. And under this condition, EL', ER' and EC' become following(this part(sereis L'C'R' circuit) is same as the part of series LCR circuit).



//ER'= -CV0*(2)

//EL'R'=EL'+ER'= -CV0*(2)/2, EL'R'+EC'=0

//Supplied energy from the primary inductor L is only EL'. And as well as the series LCR circuit of fig1, absolute quantity of thermal energy ER' is twice as much as it.
//However, EL'R' is the excess of consumed thermal energy at R'. EC' must be the absorbed energy to offset the excess of EL'R'.
//From this result, I presupposed the cause of Hutchison effect as following.

"Under the condition 1, absorption of deficient electrostatic energy will happen in the capacitive space of capacitor C', and many kinds of materials which is put in the capacitive space of C' will be melted with being absorbed its (some kind of)energy."

To make sure of this presupposition, I made comparison fig3 with Dr. Hutchison's(I conjectured as it) Tesla Coil(part of series L'C'R' circuit).

fig.4 Comparison with fig3.

//Condition of series L'C'R circuit(fig3) is " R'=2(L'/C')*(1/2), w=(1/C'L')*(1/2) ".
//In case of this setting(fig4), metalic globe and a certain domain of ground consists capacitor C', and the distance between the grounded point of L' and a certain domain of ground(colored by faded purple) becomes large insulating resistance R'. There is a close resemblance between these two circuits.
//Then, what was the cause of generating strange waveform of voltage at the primary series LCR circuit?
//Dr. Hutchison constructed this equipment to make an experiment of wirelss-transmitting of electricity. Under this condition, it is conjectured that capacitor of the primary circuit " C " had to have large electrostatic capacity because it is necessary to transmit large electric power. And the primary inductor L always has little inductance for the purpose of transforming voltage.
//These conditions make the value of "L/C" be little. When this value is too little, the resistance R which exists naturally in the LC(R) series circuit tends to become larger than 2(L/C)*(1/2) against the will of the producer.
//In consequence, it is presupposed that unexpected waveform of voltage "VC=V0e*(-wt) ("w" is actual number)" was generated in the primary LC(R) series circuit, and absorption of electrostatic energy happened at a certain place between the two secondary inductors of Tesla Coils and their metalic globes which caused to melt some kinds of materials.

//Apart from this phenominon(material's melting), the cause of melted material's floating can be presupposed as following.
//Vector of electrostatic field which is between the metalic globe and a certain domain of ground directs perpendicular at the middle point of the two secondary capacitor.
//If E1( E(y+dy,t) ) and E2( E(y,t) ) has different size( |E2| > |E1| ), dE/dy( ={E(y+dy,t) -E(y,t)}/dy ) doesn't become zero.
//Electric field which alters belong time and axis and never changes its direction lifts dielectric substance or conductive materials. It is known as Biefeld-Brown effect. I think this effect is the cause of material's floting.

//I think, to set up the insulating resistance R' properly with depending on unexpectable chance was the cause of difficulty of reappearing this pheminon. In this sense, so many accidental coincidence caused the discovering of this phenominon.
//But, if "condition 1" was setted on some TC which is constructed properly and was given enough electric energy, it may be able to reappear Hutchison effect in the capacitive space of its capacitor.