 ## More Math Non-homogenous Solutions

Adding a third-order differential term to the equations of Classical Electromagnetics (or Mechanics) increases the order of the solution as well.  For example, in attempting to solve the relevant equations, it is straight-forward to obtain an equation of the form:

W L a3 - L a2 + a R  - (1/C)  =  0

(Equation 1)

Equation 1 is a cubic equation in a (as opposed to quadratic), which can be written as:

a a3 + b a2 + c a  + d  =  0

where

a  =  1;      b  =  - 1/W ;       c = R/WL ;       d = - 1/WLC

(Equations 2)

[Note the WLC term !]

To solve Equation 2, a “reduced cubic” is used, which is defined by:

a  º  y - b/3

such that                                                                                                                    (Equation 3)

y3  +  py  + r  =  0

and

p  =  c  - b2 / 3 ;        r  =  d  -  b c / 3  +  2 b3 / 27

(Equations 4)

The solutions of the original cubic equation are then given by:

a1  =  y1  - b / 3 ;         a2  =  y2  -  b / 3 ;         a2  =  y3  -  b / 3

where

y1  =  Y1/3   +  B1/3    ;     y2  =  G Y1/3  +  G2 B1/3    ;     y3  =  G2 Y1/3   +  G B1/3

(Equations 5)

and where

Y  =  - r/2  +  {p3 /27 + r2 /4}1/2

B  =  - r/2   -  {p3 /27 + r2 /4}1/2

and

G  =  ( -1 + i Ö3) / 2   ;        G2  =  (-1 - i Ö3) / 2

(Equations 6)

M  º  {p3 /27 + r2 /4}1/2

(Equation 7)

which becomes for one special case:

M  =  {(1/W4)[1/27LC - R2/108L2] + (1/W3)[R3/27L3 - R/6L2C] + (1/W2)[1/4L2C2]}

(Equation 8)

Homogenous Solutions

The complete solution to a differential equation is comprised of both a non-homogenous solution and a homogenous solution (also known as the particular and complementary solutions).  In order to obtain this complete solution -- for both the mechanical and electrical circuit equations -- the non-homogenous solution must be added to the solution to the homogenous equation given by:

Ao X + A1 dX/dt + A2 d2X/dt2  + A3 d3X/dt3  =  0

(Equation 9)

The homogenous solution to equation 9 is of the form:

X  =  J1 e-l1t  + J2 e-l2t  +  J3 e-l3t

(Equation 10)

where the J’s are constants and l represents the roots of the characteristic equation:

a l3 + b l2 + c l  + d  =  0

and where

a  =  1;      b  =  A2/A3 ;       c = A1/A3 ;       d = A0/A3

(Equations 11)

The form of Equations 11 is the same as for Equations 2, except Equations 11 do not include the negative signs in the expressions for b and d. Equation 11 does yield for the WLRC circuit:

a  =  1;      b  =  1/W ;       c = R/WL ;       d = 1/WLC

(Equations 12)

Applying the same techniques as in Equations 3 through 8:

l1  =  z1  - b / 3 ;        l2  =  z2  -  b / 3 ;       l2  =  z3  -  b / 3

where

z1  =  Y1/3   +  B1/3    ;     z2  =  G Y1/3  +  G2 B1/3    ;     z3  =  G2 Y1/3   +  G B1/3

Y  =  - r/2  +  {p3 /27 + r2 /4}1/2

B  =  - r/2   -  {p3 /27 + r2 /4}1/2

G  =  ( -1 + i Ö3) / 2   ;        G2  =  (-1 - i Ö3) / 2

and

p  =  c  - b2 / 3 ;        r  =  d  -  b c / 3  +  2 b3 / 27

(Equations 13)

By defining:

M’  º  {p3 /27 + r2 /4}1/2

(Equation 14)

the following is obtained:

M’  =  {(1/W4)[1/27LC - R2/108L2] + (1/W3)[R3/27L3 - R/6L2C] + (1/W2)[1/4L2C2]}

(Equation 15)

This is the same as Equation 8, i.e. M = M’.

However, the roots of the cubic equations, l and a are not necessarily equal.  (The reason that M = M’ is due to the fact that the only difference in Equations 2 and 14 stems from the negative signs for both b and d.  However, these variables only appear in pairs, and thus the negative signs cancel each other out.  Thus M = M’.)

Equations 8 and 15 are important in that in the more traditional interpretations:

1)  If M (= M’) is positive, each of the roots of the cubic equation are positive and the system is considered to be overdamped.

2)  If M is negative, the roots of the cubic equation include complex roots, and the system is underdamped.

3)  In the case of M = 0, there is a real double root and the system is considered to be critically damped.

For certain special cases, the determination of the damping condition can be obtained from either the homogenous or the non homogenous solutions.

Damping

Damping is extremely important.  For a critically damped system, i.e. M = 0, Equation 8 can be rewritten in a slightly modified form:

M  =  (1/W2LC) {1/4LC - R/6WL + 1/27W2} + (R2/27W3L2){R/L - 1/4W}

(Equation 16)

In order for M = 0, in the case 1/C ¹ R ¹ 0, we must have each of the terms in brackets equal to zero.  Thus,

R/L  = 1/4W       or         W = (1/4)(L/R)

and

1/4LC  = R/6WL  -  1/27W2         or         (L/R)(RC)  =  54 W2

(Equations 17)

Equation 17, in the simplified form LC = 54 W2, could be combined with Equation 14a of Mathematical Theory (where w2  = 1 / LC for super-resonance), such that, 54 (Ww)2 = 1, or

(Ww)2 = 1/54 = (1/2)(1/27)

or

Ww  =  Ö(1/54)  =  0.136...

[For the slightly under-damped case, M is negative, such that (L/R)(RC)  >  54 W2.]

For critical damping, the system configuration must be such that W = L/4R (or W = tL/4; where tL is the inductive time constant), and W = 2tC/27 (where tC is the capacitive time constant), or

W  =  (1/2)2 tL    and    W  =  2 (1/3)3 tC

(Equations 18)

In other words, critical damping only occurs for a specific combination of W, L, R and C, while in the case of a simultaneous super-resonance, we have also that:

Ww = Ö[(1/2)(1/3)3].

(Equation 19)

We are assuming that all four variables are characteristic of the coil or circuit, but W may incorporate the inductive, resistive, and capacitive variables, and thus be the determinant -- with the frequency -- of critical damping!

From Sacred Geometry, we know that Ö2 (or Ö(1/2) ) and 27 (or Ö(1/3)3 ) are important -- including the connection to Hyperdimensional Physics and Superstrings.  It is feasible that there is a multi-dimensional connection contained within the physics of The Fifth Element.  The precise nature is only now becoming apparent.

It could be assumed that for a very large coil or very large circuit, W (and the other variables) might actually vary depending upon the amount of the coil or circuit affected by the fluctuating Emf.  In this case, W, L, R and C might have transient values different from their measured steady state values.  In effect, as the frequency of the Emf input increases, less coil or total circuit is used or affected, and thus W, L, and R apparently have decreased values (while C may appear to have increased).  The fundamental paradigm shift is that the circuit can be treated as a complete, whole circuit only if the entire circuit is affected.  Alternatively, the reaction of the circuit may involve a “center of inductance”, such that W and L might not change, but that R and C might!

In the case where the capacitance of the system can be neglected, the condition for a critically damped system is W = tL/4.  It follows that in order to obtain an under-damped condition, W > tL/4.  This requirement directly brings up the subject of time constants.

LRC Time Constant

A traditional treatment of the LRC circuit equation is:

E  =  L d2q/dt2 + R dq/dt + (1/C)q

(Equation 20)

where by assuming the non-homogenous solution: q = EC + EC(e-at), differentiating and substituting into the original equation, one can solve for the exponential constant, a.  The end result is:

a  =  { 1 ±  [ 1  - 4(L/R)(1/RC) ]1/2 } / 2(L/R)

or

TLCR  º  1 / a  =  2(tL) / { 1 ±  [ 1  - 4(tL/tC) ]1/2 }

(Equation 21)

In Equation 21, an LRC circuit time constant is shown in terms of the inductive time constant (tL) and the capacitive time constant (tC).  In the general LRC circuit, it is possible to have a complex time constant.

For example, for individual circuits where R = 2000 ohms, L = 4 henries, and C = 1 micro farad, the time constants are: tL = tC = 2 x 10-3 seconds.  This corresponds to an overall time constant of: TLCR  =  2 (tL) / { 1 ±  i Ö3 } -- true whenever tL = tC.

WLRC Time Constant

Following the same methodology and using the assumed non-homogenous solution:

q  =  Eo C  -  (Eo W/L)( e-at)

a solution for a can be obtained.  In this case, a turns out to be the roots of the cubic equation:

a3 + (-1/W) a2 + (R/WL) a  - 1/WLC  =  0

or

W a3 - a2 + (1/tL) a  - 1/tCtL  =  0

(Equation 22)

This cubic equation has already been solved for the special case.

Time Constant Implications

Modern textbook treatments of the inductive and capacitive time constants imply that the definitions are presumed to be arbitrary and in fact arise only from the mathematical anomaly involved in the exponential function.  For example, one well-accepted, classical textbook, states that, “the physical significance of the time constant... is that time at which the current in the circuit will reach a value within 1/e (about 37%) of its final equilibrium value.”  That’s it?  That’s all?

Maybe not. In fact, it is not necessary to assume that the time constants are arbitrary or trivial, but instead may represent a significant physical quantity.  It is well established in circuit theory, for example, that the sudden imposition of an applied Emf force does not result in an immediate current, where i = E /R in accordance with Ohm’s Law.

In a plot of the current versus time, when an Emf is suddenly introduced into a circuit, the area under the curve from time zero to tL will equal the area under the curve from tL to time infinity.  The fact that the areas are equal might be considered to be a simple manifestation of the mathematics and not a function of the physics.  However, there is nothing to suggest the time constant(s) may or may not have greater significance.

A more radical concept is that the area above the curve and bounded by a horizontal line indicating the final, steady-state current represents the reaction (from the universe) to the impulse Emf input as the current in the circuit approaches its steady state value.  The validity of this assumption is, of course, critical.  But interestingly enough, they can be shown to derive from the law of the conservation of energy!

The logic proceeds thus: At the time of the imposition of the voltage (particularly when the time frame is less than the time constant), one is faced with a situation where a given (and constant) voltage Eo is abruptly part of a circuit with a constant resistance, R. At the same time, however, the current i is temporarily unequal to Eo /R. In effect, for any given time frame Dt (within the time frame from t = 0 to t = ¥),

i Eo Dt  ¹  (Eo/R) Eo Dt

(Equation 23)

in contradiction to the law of the conservation of energy (or alternatively, Ohm’s Law).

However, an adherence to the law of the conservation of energy (and at the same time, keeping Ohm’s Law) can be had by requiring the universe to provide the additional energy during the time when the circuit is in the process of attempting to respond to the imposed Emf.  In effect, the universe provides an energy equal to (E2/R) Dt - i E Dt. Conservation of energy is then obtained by virtue of the fact that:

[ i E Dt ]  +  [ (E2/R) Dt - i E Dt ]  =  [ (E2/R) Dt ]

[circuit current energy]  +  [universe’s energy]  =  [steady state energy]

(Equation 24)

Note that at time zero, when the voltage is first applied, the circuit current energy is minimal, if not zero.  Because the steady state energy -- defined by the steady state condition of the voltage, resistance and current at time “infinity” -- is already determined by the steady state constants, then the universe must contribute enough energy to balance the equation!  As the current in the circuit begins to increase (at time tL > t > 0), the universe immediately begins to “back off”, decreasing its contribution to the energy equation, just as the current in the circuit responds and picks up the difference.  (The steady-state energy portion, of course, remains constant.)  By the time of t = tL, the universe’s contribution equals the circuit current’s contribution -- which, in effect, defines the time constant of the circuit.  At times t > tL, the universe’s contribution continues to decline, approaching zero as t ® ¥.

However!!!  The universe’s contribution to the system is never precisely zero!  The exponential function does reduce this contribution to an extremely small number -- and neglectable in the vast majority of steady-state conditions -- but it never quite makes it to zero.  This would therefore imply a continuing connection / interaction between the universe and the circuit (and everything else?).  This realization provides a much better understanding of the potentially supreme importance of Mach’s Principle and those other aspects of Connective Physics.

The fact that the universe’s contribution to the system always exists is extremely profound -- implying the fundamental paradigm of connection (vice separatism).  The fact that this energetic connection -- from a mathematical point of view -- might be quite small does not dismiss it.  It is, in fact, likely that when the universe’s contribution connection is on the order of Planck’s Constant, that it thereafter discontinues its rush to zero, and instead becomes a continuing connection at the Quantum Physics level.

Note also that this logic is really the same logic that began the tale of Connective Physics: The imposition of a mechanical force does not immediately result in an acceleration of the object upon which the force is being imposed.  This difference must be taken into account, and we have again assumed that the accounting is in the form of a time delay factor.  From a Conservation of Energy viewpoint, we look for a universal contribution to balance the equation for all times -- from t = 0 to t = 0.0000001 seconds to t = ¥!

Sacred Geometry

Critical to the concept of obtaining power ratios in excess of unity is the idea that the “acceleration of current” (i.e. d2i/dt2) is important and more over represents a contributing factor to the differential equations describing the electrical circuit.  [Obviously the same can be said of the mechanical system, but practical considerations strongly imply that the durability of electrons will far exceed any macro-mechanical system when the input is a high frequency, oscillating “force”.]

In those cases, where the Resistance and Capacitance can be initially neglected (as in super conductivity and a very large capacitor), we can derive equations similar to:

( -  W2 a2  +  W a   +  1 )  Eo (e-at)  =  0

Solving for a:

a = (1 ± Ö5) / (2W) = 1.618/W and - 0.618/W.

(Equation 25)

The fact the constant equals the so-called Golden Mean is undoubtedly significant.  This really cannot be chance. And just as the time constants of the circuit turned out to be important, this result is also striking.

For the more general case where R ¹ 0 (but the capacitance is still neglectable), the equation to solve is:

E  =  W L d2i/dt2 + L di/dt + R i

(Equation 26)

The non-homogeneous portion of the solution is of the form:

i   =  (Eo W/L)[ L/RW - (e-at) ]

where

a  =  (1/2W) { 1 ± [ 5 - 4W(R/L)]1/2 }

(Equation 27)

When the resistance approaches zero:

a  ®  (1/2W) ( 1 ± Ö5 )  =  F/W ; - f/W

The difference in the constant a, with and without resistance, may be important. It is possible, for example, to think of the 4W(R/L) term as a kind of “correction” to the resistance-less case of a = F/W or -f/W.  This slight variation of the “more perfect” sacred geometry value appears to be in line with one of the more notable features of the universe at large, where exact symmetries are “broken”, and such classic equations as F = m a need the addition of a “correction term” (e.g. W m d2i/dt2).

There is evidence, for example, that without the slightest of asymmetries, a perfectly round, smooth, or spherical object cannot interact with the universe!  This may help to explain, among other things, the profound importance of Transcendental Numbers, and the fact that they can not be precisely determined.

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