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Copyright Dr. Eng. Jan Pająk

Chapter F: The Oscillatory Chamber

F5. Mathematical model of the Oscillatory Chamber

Our present knowledge of magnetic and electric phenomena enables us to deduce the equations expressing the values of the resistance, inductance and capacitance of the Oscillatory Chamber. Further combination of these equations will lead to the prediction of the behaviour of this device.

This subsection is just intended to describe the Oscillatory Chamber in such language of mathematics. Therefore it supplies the vital interpretative foundations for all the researchers experimenting with this device. Unfortunately for the readers less oriented towards mathematics, it may spoil the pleasure of familiarizing themselves with the content of this monograph. For this reason, those readers who experience a revival of sleepiness each time they encounter a mathematical equation are recommended to shift from this point directly into the beginning of subsection F6.

F5.1. Resistance of the Oscillatory Chamber

The general form of the equation for the resistance of any resistor of cross-section "A" and length "l" is as follows:

R = Ω*(l/A)

In this equation the "Ω" represents the resistivity of a material from which the resistor is made. In our case it will be the maximal resistivity of the dielectric gas that fills the Oscillatory Chamber, determined for the conditions of the initial moment of electric breakdown. In turn operators "*" and "/" adopted from computer programming, mean "multiply" and "divide".

If in the above general equation, we replace the variables by the specific parameters determined for the Oscillatory Chamber, i.e. l=a and A=a2 (compare with Img.012 (F1b), this gives:

R = Ω/a (F1)

The equation (F1) received here represents the resistance of the Oscillatory Chamber, which is a function of the chamber's side wall dimension "a".

F5.2. Inductance of the Oscillatory Chamber

The determination of the chamber's inductance is an extremely difficult and complex task. Completing it with total accuracy is beyond the author's knowledge of the subject. Also a number of experts consulted in this matter were unable to help. (Perhaps some of the readers know how to resolve this problem - in such a case the author would warmly welcome a review of their deductions and the final equation they derived.) Being unable to find the exact solution, the author decided to apply temporarily a simplified one. To justify this simplification it should be stated that the deducted equation (F2) for the value of inductance will be used only once in the entire monograph, when the meaning of factor "s" (see (F5)) is interpreted. Therefore all the vital equations in this work remain unaffected.

In the simplified deductions of the chamber's inductance an assumption is made that a unitary inductance of a stream of sparks (i.e. the inductance related to the unit of a spark's length) will be equal to the inductance of the equivalent strand of wires. This assumption allows for the application of a well-known equation for the inductance of a solenoid (see the book [1F5.2] by David Halliday et al, "Fundamentals of Physics", John Willey & Sons, 1966):

L = μ*n2*l*A

When in this equation we substitute: n=p/a, l=a, and A=a2 (where "p" is the number of segments in each of the chamber's plates, whereas "a" is the dimension of the chamber's walls), the simplified equation for the inductance of the Oscillatory Chamber is derived:

L = μ*p2*a (F2)

It can be theoretically asserted that the unitary electrical inertia of a stream of sparks should be greater than such an inertia in the equivalent strand of wires. The justification for this assertion can be obtained from the analysis of the inertia mechanism. The inertia reveals itself only when the motion involves the reversible phenomena or media which absorb energy in the initial stage of the motion's development, and which release this energy when the motion declines. The greater the number of such phenomena and media involved, and the higher their energy absorption, the larger is the resultant inertia. The stream of sparks jumping through gas in every aspect manifests better potentials for causing an inertia higher than the one of a current flowing through wires. The first reason for this lies in the more efficient energy absorption and releasing by sparks, occurring because:

a) The speed of electrons in a spark can be higher than in a wire,

b) The contiguous sparks can pass closer to each other because they do not require thick insulation layers in between them (as is the case for wires).
The second reason for the higher inertia of sparks in gas results from them involving a variety of reversible phenomena - not appearing at all during flows of currents through wires. These are:

c) The ionization of surrounding gases. This, due to the returning of the absorbed energy, supports the inertia of the process at the moment of the sparks' decline.

d) The motion of heavy ions, whose mass absorbs and then releases the kinetic energy.

e) The initiation of hydrodynamic phenomena (e.g. dynamic pressure, rotation of the gas) which also will be the cause of the charges' dislocation and energy return at the moment of the sparks' decline.

The above theoretical premises should not be difficult to verify by experiments described in subsection F8.2. (e.g. stage 1c).

F5.3. Capacitance of the Oscillatory Chamber

When we use the well-known equation for the capacitance of a parallel-plate capacitor, of the form:

C = ε*A/l

(where "ε" is the dielectric constant of this capacitor, "A" is the surface area of electrodes, while "l" is the distance between these electrodes), and when we apply the substitutions: A=a5, l=a, this yields the final equation for the capacitance "C" of a cubical Oscillatory Chamber:

C = ε*a (F3)

(i.e. the capacitance "C" of a cubical Oscillatory Chamber is equal to the value of the dielectric constant "ε" for the dielectric gas that fills this chamber, multiplied by the side dimension "a" of this chamber).

F5.4. The "sparks' motivity factor" and its interpretation

Each of the equations (F1), (F2) and (F3) describes only one selected parameter of the Oscillatory Chamber. On the other hand, it would be very useful to obtain a single complex factor which would express simultaneously all electromagnetic and design characteristics of this device. Such a factor is now introduced, and will be called a "sparks' motivity factor". Its defining equation is the following:

s = p*(R/2)*√(C/L) (F4)

Notice, that after expressing this in the notation of computer languages, in which the symbol "*" means multiplication, the symbol "/" means division, the symbol "+" means addition, the symbol "-" means subtraction, while the symbol "sqrt()" means the square roof from the parameter provided in brackets "()", the above equation (F4) takes the following form:

s = p*(R/2)*sqrt(C/L).

Notice that, according to the definition, this "s" factor is dimensionless.
Independently from the above defining equation (F4), the "s" factor has also an interpretative description. This is obtained when in (F4) the variables R, L and C are substituted by the values expressed by equations (F1), (F2) and (F3). When this is done, the following interpretative equation for "s" is received:

s = (1/(2a))Ω√(ε/μ). (F5)

Notice, that after expressing this in the notation of computer languages, in which the symbol "*" means multiplication, the symbol "/" means division, the symbol "+" means addition, the symbol "-" means subtraction, while the symbol "sqrt()" means the square roof from the parameter provided in brackets "()", the above equation (F5) takes the following form:

s = (1/(2*a))Ω*sqrt(ε/μ).

Equation (F5) reveals that the "s" factor perfectly represents the current state of all environmental conditions in which the sparks occur, and which determine their course and effectiveness. It describes the type and consistency of the gas used as a dielectric, and the actual conditions under which this gas is stored. It also describes the size of the chamber. Therefore the "s" factor constitutes a perfect parameter which is able to inform exactly about the working situation existing within the chamber at any particular instant in time.

The value of the "s" factor can be controlled at the design stage and at the exploitation stage. At the design stage it is achieved by changing the size "a" of a cubical chamber. At the exploitation stage it requires the change of the pressure of a gas within the chamber or altering its composition. In both cases this influences the constants Ω, μ and ε, describing the properties of this gas. (Note that constants "Ω", "μ", and "ε", are: Ω = resistivity of a dielectric gas within the chamber determined at the moment of electric breakdown in [Ohm*meter], μ = magnetic permeability of a dielectric in [Henry/metre], ε = dielectric constant for a gas filling the chamber in [Farad/metre].)

F5.5. Condition for the oscillatory response

From the electric point of view the Oscillatory Chamber represents a typical RLC circuit. The research on Electric Networks has determined for such circuits the condition under which, once they are charged, they will maintain the oscillatory response. This condition, presented in the book [1F5.5] by Hugh H. Skilling, "Electric Network" (John Willey & Sons, 1974), takes the form:

R2 < 4*L/C

If the above relation is transformed and then its variables are substituted by the equation (F4), it takes the final form:

p > s (F6)

The above condition describes the design requirement for the number "p" of segments separated within the plates of the Oscillatory Chamber, in relation to the environmental conditions "s" existing in the area where the sparks appear. If this condition is fulfilled, the sparks produced within the Oscillatory Chamber will acquire an oscillatory character.

To interpret the condition (F6), a possible range of values taken by the factor "s" should be considered (compare with the equation (F5)).

F5.6. The period of pulsation of the chamber's field

From the RLC circuits we know that the period of their oscillations is described by the equation:



Notice, that after expressing this in the notation of computer languages, in which the symbol "*" means multiplication, the symbol "/" means division, the symbol "+" means addition, the symbol "-" means subtraction, the symbol "x**2" means "x" to the power of "2", while the symbol "sqrt()" means the square roof from the parameter provided in brackets "()", the above equation takes the following form:

T = (2*π)/(sqrt(1/(L*C) - (R/(2***2) = 2*π*sqrt(L*C/(1 - ((R**2)*C)/(4*L))).

If the defining equation (F4) on the factor "s" replaces in the above a combination of R, L, and C parameters, whereas equation (F1) and equation (F3) provide the values for R and C, then this period is described as:

(F7)

Notice, that after expressing this in the notation of computer languages, in which the symbol "*" means multiplication, the symbol "/" means division, the symbol "+" means addition, the symbol "-" means subtraction, division, while the symbol "sqrt()" means the square root from the parameter provided in brackets "()", the equation (F7) takes the following form:

T = (π*(p/s)*Ω*ε)/sqrt(1 - (s/p)**2).

The final equation (F7) not only illustrates which parameters determine the value of the period of pulsations "T" in the Oscillatory Chamber, but also shows how the value of "T" can practically be controlled. Thus this equation will be highly useful for the understanding of the amplifying control of the period "T" of field pulsation described in subsection F6.5.

If we know the period "T" of chamber's field pulsations, then we can easily determine the frequency "f" of pulsations of this field. The well known equation linking these two parameters is as follows:

f = 1/T (F8)

Of course, according to the above equation (F8), the control over the frequency "f" of the field's pulsations will be achieved via influencing the value of the period "T" of this field pulsations.

F6.
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