The Telegraph Equations
From the previous sections it has been established a set of dimensional relations as derived from four basic electrical laws,
(1) The Law of Dielectric Proportion (a) Farad, or Coulomb per Volt (b) Coulomb, or Volt – Farad (2) The Law of Magnetic Proportion (a) Henry, or Weber per Ampere (b) Weber, or Ampere – Henry (3) The Law of Magnetic Induction (a) Volt, or Weber per Second (b) Weber, or Volt – Second (4) The Law of Dielectric Induction (a) Ampere, or Coulomb per Second (b) Coulomb, or Ampere – Second
Recombination of these dimensional relations, or electrical laws, then expresses a pair of ratios, of primary dimensions in variation with respect to time. Hence, for the dielectric field,
(5) Farad per Second, or Siemens.
And for the magnetic field,
(6) Henry per Second, or Ohm.
From this pair of dimensional relations are derived a series of energy transfer and storage co-efficients. Grouping these into a pair of categories, these are given as,
(I) Energy Storage and Dissipation (a) Dielectric Energy Storage, Suceptance, B, in Farad per Second (b) Magnetic Energy Storage, Reactance, X, in Henry per Second
(c) Dielectric Energy Dissipation, Conductance, G, in Siemens (d) Magnetic Energy Dissipation, Resistance, R, in Ohm (II) The Energy Consumption or Production (a) Dielectric Energy Consumption, Conductance, G, in Farad per Second, Or Siemens (b) Magnetic Energy Consumption, Resistance, R, in Henry per Second, Or Ohm
(c) Dielectric Energy Production, Acceptance, S, in Farad per Second, Or Siemens (d) Magnetic Energy Production, Receptance, H, in Henry per Second, Or Ohm
It should be noted that these various groupings of coefficients exist in distinct, independent, time frames. The dissipation coefficients are the result of random molecular variations, that is, noise. The consumption coefficients are harmonic in nature, relating to the operating frequencies, likewise for the production coefficients. The random and the harmonic time functions are NOT ADDITIVE. In general, the combinations of these coefficients appear as versor sums. More on this later.
Since the total electric induction is the product of the total dielectric induction and the total magnetic induction, there exists the products of the coefficients of dielectric induction and the coefficients of magnetic induction. These products give rise to a set of electrical factors. These factors, the product of the dielectric part, in Siemens, and of the magnetic part, in Ohm, gives rise to the dimensional relation
(7) Ohm – Seimens, or Numeric
Hence, this derived dimensional relation, or FACTOR, is a numeric, that is, dimensionless. Since both the Ohm and the Siemens are versor quantities, it follows that this numeric is also a versor, a dimensionless versor magnitude. It is not a scalar, it is a versor with a position in time.
These factors are hereby established to be dimensionless versor magnitudes. Combining the dielectric and magnetic coefficients gives the following factors,
(a) The Energy Storage Factor, XB, in Ohm – Siemens, or Henry – Farad per Second Squared (b) The Energy Loss Factor, RG, in Ohm – Siemens (c) The Energy Gain Factor, HS, in Ohm – Siemens, Or Henry – Farad per Second Squared.
Hereby it is, HS supplies the energy, XB holds the energy, RG removes the energy. These three factors define the movement of electricity thru the dimension of time, this for a generalized electrical configuration.
It is usually that the electrical configuration, the metallic-dielectric geometry, exhibits only energy losses, no component of energy gain exists. An example is one span of a J-Carrier open wire transmission line. This line holds energy in its bound electric field of induction, but a portion of this energy is lost thru molecular action within the glass insulators and within the copperweld wires. There exists no component of energy gain in this span of open wire line. Here it is the parametric terms vanish. No factor HS exists and RG is pure dissipation. These simplifications allow for the algebraic expression in an archetypical form of the generalized electrical configuration.
Given the basic dimensional relations,
X, the Reactance, in Henry per Second
B, the Suceptance, in Farad per Second
These relations representing energy exchange between the dielectric field, and the magnetic field, of inductions. This energy exchange is in an alternating form. It is also
R, the Resistance, in Ohm
G, the Conductance, in Siemens
These relations representing energy removal from the magnetic field, and the dielectric field, of inductions. This energy loss is in a continuous form. Hereby XB is the alternating “current” factor, and RG is the direct, or continuous, “current” factor.
Obviously, in the situation of an electric generator, HS could replace RG in such a configuration. Here energy is produced in a manner of negligible losses, and thus RG drops out of the equation. It is however, a system or configuration exhibiting both loss and gain requires a more complex algebraic expression. This is developed in the final section of “Symbolic Representation of the Generalized Electric Wave” by E. P. Dollard.
Combining terms with like dimensional relations, that is, Ohm and Henry per Second, or Siemens and Farad per Second, gives rise to a total impedance, or a total admittance of the electrical configuration. Hence it is,
(I) The Total Admittance, Y, in Siemens (8) Y = G – jB The versor sum of the Conductance, G, in Siemems, and the Suceptance, B, in Farad per Second. (II) The Total Impedance, Z, in Ohm (9) Z = R + jX The versor sum of the Resistance, R, in Ohm, and the Reactance, X, in Henry per Second.
Here Y represents the dielectric field, and Z represents the magnetic field.
The electric field is the product of the dielectric field, and the magnetic field. Q is Psi times Phi. Taking then the product of the total dielectric Admittance, Y, in Siemens, and the total magnetic Impedance, Z, in Ohm, gives the dimensional relation
(10) Siemem – Ohm Or Numeric
Hence ZY is a dimensionless magnitude, it having a versor position in time, since both Z and Y have a versor position in time. The product of the two versors is also a versor. ZY is not scalar, it is a dimensionless versor magnitude. It represents a wave propagation in the dimension of time, a TIME WAVE.
It has been given that the product ZY, in Siemens – Ohm, is a dimensionless magnitude having a versor position in time. It is the product of a pair of versor sums
Y = G – jB Z = R + jX
However, it is the product of versor sums are also versor sums. Taking this product
ZY = (R + jX)(G – jB)
And factoring like terms, gives the following factors,
(I) The POWER FACTOR a = (XB + RG)
Where a is the power factor, the percent of energy lost from the total movement of electrical energy in an electrical configuration.
(II) The INDUCTION FACTOR b = (XG – RB)
Where b is the induction factor, the percent of energy stored by the total movement of energy in an electrical configuration.
Here the resulting sub-factors are defined, for the power factor,
XB, the factor representing the cyclic exchange of energy between dielectric and magnetic forms. RG, the factor representing the acyclic dissipation of energy from both dielectric and magnetic forms.
And, for the induction factor,
XG, the factor representing the transfer of energy out of magnetic form and into dielectric form. RB, the factor representing the transfer of energy out of dielectric form and into magnetic form.
The propagation constant, ZY, in Ohm – Siemens can hereby be expressed as the versor sum of the power factor, a, in percent, and the induction factor, b, in percent. This results in,
ZY = ha +jb Ohm – Siemens, or Total Percent
ZY must always equal 100 percent but it has a variable position in time, this expressed as a resultant of ha and jb.
The versor operators are defined as,
h, the roots of the square root of positive one j, the roots of the square root of negative one
Expanding the expression for the propagation constant, ZY, as a versor sum of the expressions for a and for b, gives
ZY = h(XB + RG) + j(XG – RB)
Hereby established is the most important algebraic expression of dimensional relations, this defining the movement in time of electrical energy in any electrical configuration.
This algebraic expression is called the “Heaviside Telegraph Equation.” It is in this expression the electrical energy is expressed directly in its four pole archetype. Note that this four polar archetype underlies all Native American artforms. Is this related to America as the birthplace electrical technology thru Tesla, Edison, and Steinmetz? Europe was too consumed in self edification mathematics, except GÖTHE.
This algebraic expression gives a pair of waves in motion thru the dimension of time, one moving forward in time, the other backward in time. See “Theory and Calculation of Transient Electric Phenomena”, C. P. Steinmetz, the chapter, “Resistance, Inductance, and Capacity”. Here (R + S) is forward in time and (R – S) is backwards in time.
Geometrically, this expression represents a pair of counter propagating logarithmic spirals. This spiral form is demonstrated in Ernst Guillimen, “Communication Networks” Volume One, and in “Theory and Calculation of A.C. Phenomena”, Appendix – “Oscillating Currents”, C. P. Steinmetz. It is important to remember that these “motions in time” are of a versor form, finding no equivalence in spatial representation except by analogy. There is no such thing as a surface of time, no 2D time. These are versor, not vector expressions.
Expressing the four distinct sub-factors, the versor combination of which gives the propagation constant, ZY, in Ohm – Siemens, it is,
(I) The Power Factor Pair
XB, the “axial” product, the longitudinal component of energy motion in time, forward and backward in an alternating manner.
RG, the “dot” product, the scalar component of energy dissipation, this independent of time.
(II) The Induction Factor Pair
XG, the “cross” product, the transverse component energy transfer thru time from magnetic to dielectric. This is a clockwise versor around axis XB.
RB, the “cross” product, the transverse component of energy transfer thru time from dielectric to magnetic. This is a counter-clockwise versor around axis XB.
In the pendantic, mystic, and dis-information world there are two products, the dot and the cross, here exists four products, axial, dot, and a conjugate pair of cross products. Here is why misunderstanding exists, the basis for the “longitudinal scalar” idiots.
In its versor form the Telegraph Equation is expressed symbolically as
k(ZY) = ha + jb
Where the magnitude, ZY, represents the electricity, and the operator, k, represents its versor position in time. This is given for a 360 degree scale on a power factor meter, an analog computer for expression of k(ZY). (ZY is the pointer, k is the scale, that simple.)
Expressing a versor relation as,
k = jh
That is, negative one to the one half power times positive one to the one half power, gives negative one to the one fourth power. Since this “fourth root” of negative one suggests a conjugate of the fourth root of positive one, it is then
k, the roots of the EIGHTH root of positive one
This as the most general versor operator for the Telegraph Equation.
Here we are beyond the scope of this elementary series of discussions on the rudiments of electrical theory. This series concludes here. But it was fun, don’t you think? It will make you think. For a more in depth study of this subject see the following;
“Theory and Calculation of A.C. Phenomena”, C. P. Steinmetz, the chapters, “Power and Double Frequency Quantities” and the appendix “Roots of the Unit”.
“Symbolic Representation” Papers by E. P. Dollard, and all references given in these papers.
“Electro-Magnetic Theory”, O. Heaviside, in particular the development of his “Telegraph Equation”.
“Physics and Mathematics in Electrical Communication”, James Owen Perinne.
Finally, for an excellent musical portrayal of the ZY relationship listen to G. F. Handel, “Alexander’s Feast, or the Power of Music”, the final coral movement. It is a good ending to this series of writings.
73 DE N6KPH SK.