2019-07-27

Spherical generator on ESCs. Calculation

In the previous part of this work we have introduced a new model and met with the math calculations for circuits with electrostatic capacitor (ESC). The missing element to the beginning of this calculation is a single coefficient \(k_s\), we find for ESCOs in the form of a spherical capacitor. It is the name of this type of capacity and determined the name of the generator, the results of the calculation which we then present for a range of initial parameters.

Find \(k_s\) for a spherical capacitor

Find the ratio \(k_s = C_1/C_S\), which, in essence, binds dooblydoo and private capacity, and therefore can be different for different types of condensers (for more on this read here). We derive this ratio for the spherical condenser (Fig. 4) which has inner conducting sphere with radius \(R_1\) and the outer radius \(R_2\).

Between fields is a space filled with a substance with a relative permittivity \(\varepsilon\), which we take equal to unity [1]. Then its capacity will be well-known formula [2]: \[C_S = {4\pi \varepsilon_0 \over \frac{1}{R_1} - \frac{1}{R_2}} \qquad (3.1)\] where: \(\varepsilon_0\) is the absolute permittivity. Also, we know the classical formula to calculate the capacity of the private sector [2]: \[C_1 = {4\pi \varepsilon_0 R_1} \qquad (3.2)\] \[C_2 = {4\pi \varepsilon_0 R_2} \qquad (3.3)\] Remembering that \(k = C_2/C_1\) and \(k_s = C_1/C_S\), find the desired coefficient for a spherical capacitor: \[k_s = {k \over k - 1} \qquad (3.4)\] Now we have all data to calculate, at least for spherical esque. To manually perform such a calculation is problematic, but math editors will cope with this task very easily, what we will do with them next.

Fig.4. The spherical capacitor |

The results of the calculation of the spherical generator

The results are presented in mathematical editor MathCAD for formulas of the second part of this work, which include the coefficient of \(k_s\) for ESCOs in the form of a spherical capacitor from the formula (3.4). Parameters inductive resistances are taken: \[X_{L1} = \Bbb{i} r_1 {0.333 \over \Delta}, \quad X_{L2} = \Bbb{i} r_1 {0.1085 \over \Delta} \qquad (3.5)\] the resistance \(r_1 = 1\) (Om). In reality, it represents the internal resistance of the generator in the amount of active resistance of coil wire \(L_1\). The input voltage of the generator \(E = 10\) (In) (see Fig. 3d). The ratio between the two capacitances is taken such that \(k = 6\).

Interestingly, the initial parameters are selected for values of the resonance, located near a classic, but not coinciding with it. Such a resonance can be in a separate category and call it __electrostatic__. It differs from the classical fact that for resonance requires selection of not two but three parameters: capacitance, inductance and resistance, and appearance, it resembles the self-excitation scheme.

Fig.5. The dependence of the balance of power between \(\Delta\) and \(k_r\) | Fig.6. The output power versus \(\Delta\) and \(k_r\) |

Figure (5) presents a plot of efficiency of the second kind (or COP) depending on \(\Delta\) and \(k_r\), calculated according to the formula (2.7) from the previous part. Figure (6), on the Y-axis is the efficiency provided directly to the power output of the generator in Watts, calculated according to the formula (2.8).

Figure (7) and (8) is the dependence of the stress \(U_1\) and \(U_2\) from the same options. Here it is clearly seen that positive feedback contributes to the buildup of these stresses, and hence an additional increase, as the parameter of efficiency and output power. Also, according to this chart it is possible to estimate the protection against overvoltage and breakdown periods, for example, between the two spheres esque.

Just for comparison, we presented the schedule for the ordinary efficiency case, if you remove the electrostatic effect \(U_g=0\) and calculate the generator without it (Fig. 9).

How to use charts

You must first choose the resistance coefficient \(k_r\) and operating point on the first graph (Fig. 5). For example, we chose \(k_r = 10^2\) is the purple graph (second from left). We find it high, from which a vertically down-delayed direct and find parameter \(\Delta\). For purple graphics it will be \(\Delta \approx 1.9\cdot 10^{-4}\). On the basis of the found parameter in the graphs is possible to estimate efficiency, output power and voltage at the test points.

Now you need to decide the working frequency of \(f\) and recalculate it in the corner: \[\omega = 2\pi f \qquad (3.6)\] to Find the remaining parameters can now be this: \[r_2 = r_1 k_r \qquad (3.7)\] \[C_1 = {\Delta \over \omega r_1}, \quad C_2 = C_1 k\] \[L_1 = {k_{L1} \over \omega^2 C_1}, \quad L_2 = {k_{L2} \over \omega^2 C_1}\] where the unknown coefficients are taken from the numerator in equation (3.5). In our case they are: \(k_{L1} = 0.333\), \(k_{L2} = 0.1085\).

__Example.__

Choose the frequency \(f = 1\) (MHz). Then the remaining parameters will be: \[r_2 = 100\, (Omh), \quad \omega = 6.28\cdot 10^6\] \[C_1 = 30\, (pF), \quad C_2 = 180\, (pF)\] \[L_1 = 282\, (\mu H), \quad L_2 = 94\, (\mu H)\] on the Basis of obtained data it is possible to calculate the radii of two spheres according to equations (3.2-3.3), and design parameters of the coils in specialized calculators.

Other initial parameters

The above example is for a single combination of initial parameters: \(r_1, k, X_{L1}, X_{L2}\). The calculation of the second part suggests plenty of options. Below is another example with other initial parameters: \[X_{L1} = \Bbb{i} r_1 {0.1 \over \Delta}, \quad X_{L2} = \Bbb{i} r_1 {0.2181 \over \Delta}\] the resistance \(r_1 = 1\) (Ω) and the input voltage of the generator \(E = 10\) (B). The ratio between the two tanks is: \(k = 2\) (the relative distance between the spheres is less than in the previous example).

Deviations from the practice

Of course, any model is only an approximation to reality. In our case, the obvious next approximation, which can affect the practical result:

- in the model in figure (3d) is not considered self-capacitance of inductors, which can be substantial, if not to accept special measures. It will change the real resonance frequency;
- calculations of power output and efficiency are made in the approximation that \(L_2\) does not have its own active resistance, which in practice will reduce these parameters;
- in the calculations is not taken into account, the efficiency of the oscillator (\(E\) in figure 3d), which in reality would proportionally reduce the overall efficiency of the entire device.

Pass effect

Why is described in this work, electrostatic effect, opened A. Volta, finds himself only occasionally? Here may be several reasons. First, it occurs in quite narrow frequency bands. And this despite the fact that other relations, for example, soglasovat resistance has already been fine-tuned; this we can see from the above charts. Secondly, only certain relationships between the schema elements. A deviation of even a few percent of them reduces this effect to zero. In the third, in some cases, even if all are finely tuned, the effect is difficult to detect due to the low output power. Ie a COP can be great, and \(P_2\) is very small. The experimenter, in this case, is focused on the power output. In many cases, positive feedback is manifested in the reduction of consumption from the power supply, not increase, \(P_2\). Fourth, due to the large jump in voltage (see graphs), the experimental scheme may fail in the form of breakdown of the spark gap or of a failure of the electronic components. Additional reasons described in the previous section. All together leaves the researcher with the incredibly narrow corridor of possibilities, but we hope that this work will allow will come nearer to it closely.

Other types of generators

In this paper, we gave a calculation of the generator for ESK on the spherical capacitor. In General, ESCs consists of two or more alternating plates, the shape of which can be, in principle, any. Example: a coaxial capacitor. Another example: a plate capacitor, consisting of two or more parallel plates (Fig. 1d). A practical option is the ESK in the form of the inductive capacitor, where two classic or bifilar wound on top of each other or next to each other, the coils serve as both inductances and capacitances for the circuit in figure (3d). A well-proven device Nikola Tesla, in which the first contour (Fig. 3d) is a transmitter, Tesla Transformer (TT) with the scope on the hot end, and the second receiving — one or more like TT. In this scheme it is necessary to consider the wave properties of the coils and the coupling coefficient \(k_s\) — count between the two spheres located at a certain distance.

__The materials used__

- Wikipedia. Dielectric permeability.
- Wikipedia. The electric capacity.