natural_asymmetry_in_electrical_systems

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A Natural Asymmetry in Electrical Systems with Far-Reaching Consequences

This text demonstrates unexplored far-reaching consequences from some naturally existingasymmetries in electricity. One such discrepancy is the disappearance of current offset even whenthere is offset in the voltage applied on a resistor and capacitor connected in series. Evidence ispresented proving that when the average slope of the theoretical energy-time dependence (theaverage power) is calculated accurately, voltage offsets exist whereby the input power becomesnot only practically zero but can be of negative value despite the current flowing through the

circuit. In other words, it is a standard, albeit overlooked, electrical phenomenon that by justsimply adjusting the voltage offset not only can energy spent per unit time be made less than theenergy produced per unit time but the input energy can be made to practically vanish and evenbe negative.

It has been established that constructions existwhich allow for a spontaneous displacement underthe action of a conservative force. This means that itis possible to obtain energy without depleting a pre-existing energy reservoir. There has been experi-mental evidence showing the production of more en-ergy than the energy spent in some electric circuits.

The apparent cause for such disbalance could be thatunder certain conditions part of the input energy thatwould match the output energy? , is saved.

Here we show that such possibility is a phe-nomenon inherent in the essence of electricity.

The function expressing the input energy at anytime τ due to flowing of current I in driven by voltageV in is

E (τ ) =

τ

0

I inV indt =

τ

0

A sin(ωt + Θ)(F + V m sin(ωt))dt (1)

The first derivative (slope) of that function at a givenpoint τ within the interval [0, T ] is

E (τ ) =d

dτ

τ

0

I inV indt = I inV in|t=τ =

A sin(ωτ + Θ)(F + V m sin(ωτ )) (2)

where V m is the amplitude of the applied voltage, V;F is the offset voltage, V; ω = 2πf is the angularvelocity, rad s−1; f = 1

T is the frequency, Hz; T is

the period, s; t is the time, s; A = V m R2+( 1

2πfC )2

and

Θ = arctan

1R2πf C

.

Therefore, the average value of the first deriva-tive (slope) of that function within the entire interval[0, T ], which constitutes the average power within this

interval, is

P from integralin =

1

T

T

0

E (t)dt =

1

T

T

0

A sin(ωt + Θ)(F + V m sin(ωt))dt =

1

T

A sin(ωt + Θ)(F + V m sin(ωt))|T 0

=

AV mcos(Θ)

2= const (3)

Alternatively, the integration can be done even by firstcarrying out the integration numerically using Simp-son’s rule and then applying the Riemann way of solv-ing the integral. Thus, for 1000 element discretizationit will be

P from integralin =

1

T

T

0

A sin(ωt+Θ)(F +V m sin(ωt))dt =@1003− @3

T (4)

where @1003 and @3 are the values of the integralin the spreadsheet cell 1003 and 3 in the column @containing the results from the integration.

On the other hand, since, as seen, A sin(ωτ +Θ)(F +V m sin(ωτ )) is the slope of the function E (t) at timeτ the average slope (the average power) within theinterval [0, T ] can be expressed as:

P from seriesin =

limn→∞ 1

n + 1

ni=0

Asin

ωi

T

n + Θ

F + V msin

ωi

T

n

(

As is known, the value of the integral (eq.(3)) is thelimit of the corresponding Riemann sum, while series(eq.(5)) is not a Riemann sum, although it expressesthe same thing as (eq.(3)) (respectively, eq.(4)). Never-theless, it is expected that eq.(3) (respectively, eq.(4))and eq.(5) should produce the same result which, in-deed, they do when the offset voltage is F = 0. How-ever, contrary to the expectation, when the offset volt-age is F = 0 , unlike the integral, eq.(3) and eq.(4), the



30

series, eq.(5) show a result which is dependent on F .This can only be checked numerically since there

is no analytical way to check what the series (eq.(5))converges to. Numerical methods are based inher-ently on partitioning the studied interval [0, T ]. How-ever, when considering numerical calculation, oneshould note that too small a partition would lead to

too rough an approximation while too large a parti-tion will lead to greater rounding errors as well asfloating point errors. Therefore, a partition P = 1000is chosen as a compromise.

It can easily be seen by using a standard spread-sheet program that when numerical integration iscarried out for P = 1000 both for offset F = 0 and foroffset F = 0, the result, as was already noted, is prac-tically constant for all values of F and is practically

equal to the value AV mcos(Θ)2 = const of the integral in

eq.(3).The more accurate numerical calculation, namely

that of the series in eq.(5), for the same P = 1000 and

offset F = 0 also gives as a result a value practicallyequal to AV mcos(Θ)

2 = const.However, when that more accurate numerical cal-

culation, namely, the numerical calculation of the se-ries in eq.(5), is carried out for the same P = 1000but the offset now is F = 0, then the result becomesa function of the offset F . For values of F > 0 theseries becomes more and more positive with the in-crease of F . The opposite is observed when F < 0.For values of F < 0 not only is the series tendingtowards zero, but after a certain F value it becomesnegative. The negative value of power means thatpower is returned to the source despite the fact that

there is current flowing through the circuit generat-ing positive output power I 2R.

Of course, there are physical limits to the de-crease (respectively increase) of F . However, the ob-

served dramatic effects in changing the P from seriesin

value, (eq.(5)), compared to the constant value of the

P from integralin , (eq.(3)), and, most importantly, com-

pared to the output power I 2R, are observed evenat modest physically viable values of V m and F , onthe order of volts.

Discussing P out

Although the point of this paper is already madewe may also observe the output power both from

integral, P from integralout and from series P from series

out .We can express these quantities in the same way wedid for the input.

The function expressing the output energy at anytime τ due to flowing of current I in through the re-sistance R is

E (τ ) =

τ

0

I 2inRdt =

τ

0

A2 sin2(ωt + Θ)dt (6)

The first derivative (slope) of that function at a givenpoint τ within the interval [0, T ] is

E (τ ) =d

dτ

τ

0

I 2inRdt = I 2inR

t=τ = A2 sin2(ωτ +Θ))

(7)

Therefore, the average value of the first derivative(slope) of that function within the entire interval [0, T ],which constitutes the average power within this inter-val, is

P from integralout =

1

T

T

0

E (t)dt =

1

T

T

0

A2 sin2(ωt + Θ)dt =

RA2

2= const (8)

Alternatively, as when calculating P in

P from integralout =

1

T

T

0

RA2sin2 (ωt + Θ) dt =

@1003 − @3

T (9)

On the other hand, since, as seen, A2 sin2(ωτ + Θ) isthe slope of the function E (t) at time τ the averageslope (the average power) within the interval [0, T ]can be expressed as:

P from series

out

= limn→∞

1

n + 1

n

i=0

RA2sin2ωiT

n+ Θ

(10)The values of eq.(10) can be calculated numericallyfor the given R, C , V m, ω and Θ. For accuracy, thevalue of the momentary (P in)t = (I inV in)t and (P out)t

= (I 2inR)t must be calculated individually, point bypoint, for every t within [0, T ] at the chosen level ofdiscretization (the higher the level of discretizationthe higher the accuracy) and the results averagedover the interval.

Conclusions Pertaining to Experiments

It should be noted that experimentally, providedthe current is measured with a Hall effect currentprobe which would ensure high accuracy and thevalue of the Ohmic resistance and the capacitance arewell known, the correct measurement of voltage is ofcrucial importance if one is to answer positively as towhether or not the data shows the effect discussedabove. At that, it is not so much the amplitude of thevoltage as is the voltage phase shift with respect tothe current that has to be measured with utmost accu-racy. It can be seen from the theoretical calculations



31

described above that while the percent difference inthe measured amplitude, which is well within the er-ror limits of a typical 8-bit oscilloscope, compared tothe actual amplitude, will only give ostensible P out inexcess of P in no more than just percents, the samepercent inaccuracy in voltage phase shift would causeostensible P out in excess of P in, already on the order

of times.Now that the reality of excess energy production

has been established definitively, based on the veryessence of the electric phenomena, one may suggestthat it very well may be that some of the observedexperimental effects when offset F = 0 could be dueto experimental errors. The origin of these errors, ifat all, can now be clearly identified. One possible ex-perimental error may be due to the phase shift whichthe voltage probe measures in its own circuit, whichmay differ and yet be attributed to the circuit understudy, as if it is the actual phase shift of V with re-spect to I in that circuit under study.

On the other hand, even for offset value F = 0 (nooffset, that is) factors external to the RC circuit suchas the transformer core or a way of winding the coils(double-wound, for instance) may cause the phase be-tween I and V to acquire values in the studied circuit

different from Θ = arctan

1R2πf C

, corresponding

to the values of the RC circuit components. Thus,the new phase shift, different from the phase shift ex-pected for the concrete R and C values of the studiedcircuit, can lead to a dramatic effect expressing itselfin the obtainment of true excess energy. Such gen-uine excess energy even at F = 0 has been observedexperimentally in a number of cases, in transformers,coils with cores, air coils, double-wound coils etc. Itis felt, however, that more precise measurements areneeded to protect these findings from being vulner-able to criticism which may dilute the firm findingspresented in this paper, conclusively proving that thepossibility to produce excess energy is inherent in theelectric phenomena.

Discussion

As can be seen from this analysis, the possibilityto obtain more energy than the energy spent is in-herent in electricity phenomena and it should be ex-plored along these lines. As seen, the excess energyin the output when F = 0 (the slight excess energyseen in Table I even when F = 0item 7 comparedto item 3is not even mentioned here in view of thedramatic excess energy values that can be obtainedwhen F < 0) is obtained due to saving of energy fromthe input. This is exactly what happens in ’cold fu-sion’, electrochemically. Here we have another sofar unnoticed example of this way of producing ex-cess energy due to the discrepancies or asymmetries

TABLE I: Numerical solution of the integrals and the series

Item Quantity Value∗∗ Ref.

1 P in =AV mcos(Θ)

22.26969283346722×10

−6 eq.(3)

2 P in =@1003−@3

T 2.26969283348716×10

−6 eq.(4)

3 P

AVERAGE(@3:@1003)∗

in 2.26742540805888×10

−6

eq.(5)4 P

AVERAGE(@3:@1003)in

3.06451263274804×10−11 eq.(5)

5 P out =RA

2

2 2.26969283346724×10−6 eq.(8

6 P out =@1003−@3

T 2.26969283346722×10

−6 eq.(9)

7 P AVERAGE(@1003:@3)∗

out2.27196010963275×10

−6 eq.(10)

8 P AVERAGE(@1003:@3)out 2.27196010963275×10

−6 eq.(10)

∗no offset; i.e. F = 0∗∗raw data from spreadsheet

inherent in the phenomena.

It should also be noted that the production of en-ergy in excess of the energy spent observed herecan be explained away only if one proves that whilespreadsheet programs such as Excel give correct re-sults at every other possible calculation it suddenlystarts to be inaccurate when the offset F in eq.(5) be-comes non-zero. Obviously, it is untenable to suggestthat such proof can be presented and therefore thisstudy adds to the categorical proof based on analy-sis of the magnetic propulsor? ,? and electrolysis inan un-divided cell? that it is possible to violate theprinciple of conservation of energy.

Numerical Example

Here are the formuli to check the above conclu-sions directly in a spreadsheet such as Excel basedon a sample physically consistent set of values. Thesevalues could be as follows: resistance R = 9.9244Ω,capacitance C = 115pF, frequency f = 800kHz, volt-age amplitude V m = 1.17V and voltage offset F =−3.356V . The current values are in column B of thespreadsheet

=(1.17/SQRT(9.9244^2+(1/(2*PI()*800000*

0.000000000115))^2))*SIN((2*PI()*800000*A3)+

ATAN(1/(9.9244*2*PI()*800000*0.000000000115)))

and the voltage values are in column C

=-3.356+(1.17*SIN((2*PI()*800000*A3)))

for times starting at time t = 0 in cell A3 with an incre-ment of 1.25×10−9s pasted up to cell A1003. Each cellin column D calculates the momentary input power,P momentary

in . This, for instance is the calculation incell D3

=B3*C3



32

while each cell in column E calculates the momentaryoutput power, P momentary

out . This, for example, is thecalculation in cell E3

=B3*B3*9.9244

The cells in F and G columns calculate the input andoutput energy up to the corresponding time, calcu-lated after the the integration of the momentary val-ues of P momentary

in in column D and P momentaryout in

column E, respectively. This is an example of suchcalculation done in cells F3 and G3, respectively

=I2+(ABS((A4)-(A3))*(((D3)+(D2))/2))

=J2+(ABS((A4)-(A3))*(((G3)+(G2))/2))

The H4 and I4 cells calculate back the value of theRiemann integral from that last and the first value ofthe energy.

=(F1003-F3)/(A1003-A3)

=(G1003-G3)/(A1003-A3)

On the other hand, cells H5 and I5 calculate the aver-age value of all the momentary power values in the

cells of columns D and E, respectively, thus producingthe value of the average input and output power.

=AVERAGE(D3:D1003)

=AVERAGE(E3:E1003)

At the end the ratios = I 4/H 4 and = I 5/H 5 can becalculated in cells J4 and J5, respectively, of the P out

P incalculated via integration as well as through the se-ries. The values obtained are presented in Table I

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