Método de evaluación de los cuestionarios

From the equation 5.32 it can be seen that the largest integral of f(x) will result in the lowest required value for E0 and thus the minimum voltage required for partial discharge.

From the graph in Figure 5.18 it can be seen the equation that would provide the largest integral is Q(x)max =QT/x. Performing this integral gives;

Z d

0

f(x)dx= 1 +ln(d

λ) (5.35)

Substituting equation 5.35 into equation 5.34 gives;

E0 =

4πρσe

1 +ln d_λ·3meλ (5.36)

5.4.0.6 Voltage Required for Partial Discharge Using Two Parallel Plates

Now E0 is the E-field that would be between the two plates should no electrolyte be present. This is related by the equation V = E0d. Therefore V(partial discharge) (Vpd) is given by;

Vpd =

4πρσed

3meλ· 1 +ln d_λ

(5.37)

Incorporating equation 5.1 gives an equation for the minimum voltage required for partial discharge; Vpd = 4πρσ 2_epd 3mekT · 1 +ln kT_σ +ln(pd) (5.38) As it can be seen from the above equation there are no curve fitting constants, all of which are natural constants and the only variable pair is ’pd’. Here we have a derivation

theoretically linking the pd product to the voltage of partial discharge. Applying the constants below

e= 1.602176487×10−19 C me= 9.10938215×10−31 Kg k = 1.3086504×10−23 ρ= 1.2041 Kgm−3 T = 300 K σ =πr_i2 = 2.82743×10−21 m2 where ri = 0.3×10−10 m gives Vpdmin = 193.3pd 0.6182 +ln(pd) (5.39)

The equation 5.38 provides the general equation for the minimum voltage that would result in partial discharge between two electrodes separated by a gaseous dielectric. The variables that correspond to the partial discharge voltage are the pd product, Tempera- ture (T), the density of the gasρand the ionic radius of the particular gas (i.e. σ =πri2).

Provided the conditions are stable for a given set up, the only variable will be the pd

product as experimentally measured by Paschen [18]. The implications of this are that for a given safe operating air gap d at for example sea level (i.e p ≈ 100KP a), a much larger air gap would be required at 15Km (50,000ft) where p≈10KP a, to maintain the same safe pd product. In this example to maintain the same safe operating parameters the air gap would have to be ten times greater than the original to operate safely at these altitudes.

Applying reasonable parameters for temperature (i.e. 300K), the density of air (i.e. 1.2041Kgm−3) and applying the parameters for the Nitrogen content in air (i.e. ri =

30nm) then the particular solution to equation 5.38 can be found in equation 5.39, this equation is graphically presented above in figure 5.19. This curve appears to have the characteristics of those experimentally measured by Paschen, i.e. at large values of pd

product, large voltages would be required to cause partial discharge, however, as the pd

product is reduced so to does the partial discharge voltage. The partial discharge voltage andpdproduct continue along these lines until a point when the curve begins to rise again for any further reduction in pdproduct and as a result the curve becomes non-monotonic. A reason for the curve being non-monotonic for low values of pd product is that the gap distance d is less than the mean free path λ of the gas. Having the condition λ < d

partial discharge rises, trending towards infinity.

In chapter 7 the predicted curve shown above in figure 5.19 will be compared against experimental results.

6

Arc

The previous two chapters have detailed a theory of how a plasma is formed, how it is able to progress and migrate from one electrode to the other. The theory, however, only details the plasma formation up to one mean free path (λ) away from the Anode. The reason why the plasma is not considered right up to and including the anode is because the theory would break down at this point. [51] [52]

The theory is still complete as the extra forces to allow the electrons to bridge the final gap (λ) have been accounted for. But once the electrons do bridge the gap and allow a conductive path to form, the forces that are required for the plasma to exist and progress will collapse, the plasma will be broken and the whole cycle will then repeat. This may account for the clicking sounds one hears prior to when an arc establishes. [53]

6.1

The final mean free path

λ

Figure 6.1: A depiction of the formation of an arc

From the above figure the plasma formation can be seen, during this phase the supply voltage is also the voltage across the air gap, with no current flowing. The plasma extends to the anode, at this point the voltage across the gap reduces to zero and the current rises and is only limited by the circuit impedance. With the voltage at zero there will be no E-field to progress the plasma so it will disperse. and the whole cycle will begin again. [54]

The question is what new phenomena may be occurring over this finalλ?

Referring to figure 5.19 one way to think about an arbitrary point on the curve is that that voltage is capable of producing a plasma that can extrude over the entire gap sepa- rating the cathode and anode electrode. If the distance of gap is considered as multiples

of λ, meaning the gapd=N λ. Now consider a slightly larger gap by one more mean free path, a slightly higher voltage would be capable of extruding a plasma a slightly further distance may be (N + 1)λ, as shown below. [55]

Figure 6.2: General change in discharge voltage with respect to ’pd’

If we therefore differentiate equation 5.38 with respect to ’pd’ and then equate d toλ. Then some insight into this region may be gained.

Performing this differentiation as described above yields the following result.

dVpd

dpλ = 0 (6.1)

At first this result was a surprise, however, it need not have been as the plasma for- mation is effectively akin to an extrusion and if the plasma is broken by a distanceλthen one would expect that the plasma fill the gap, thus requiring no change in voltage.

Figure 6.3: A depiction of the plasma with a gap onλ

Referring back again to figure 6.1, the top two captions depict a binary state of partial discharge, whereby either V gap = 0 or Igap = 0. However, is there a parameter that could vary that would enable the state where Vgap and Igap are both non-zero? [56] [57]