The Interruption of the Diffuse Vacuum Arc for ac Currents Less Than 2 kA (rms.) with a Fully Open Contact Gap
In this section, a fully open contact gap can mean a contact gap of at least several millimeters. At current zero in an ac circuit the rate of change of current is given by:
FIGURE 4.3 The transient recovery voltage (TRV) showing three ways of measuring the rate of rise of recovery voltage (RRRV).
i.e.,
For ас currents of about 2000A (rms.), the rate of change of current at current zero is thus approximately 1 A/ps. We already know from Section 2.3.3 in this volume that the cathode spots in a diffuse vacuum arc can react almost instantaneously to changes in current up to ЮОА/ps. Thus, as the current approaches current zero, these vacuum arcs will be characterized by a gradual extinction of the cathode spots, until just before current zero when only one spot remains. Also, from the discussion of current chop in Sections 2.3.3 and 3.2.6 in this volume, this final cathode spot will selfextinguish just before the true current zero. As I discussed in Chapter 2, the anode contact during the arcing phase of the diffuse vacuum arc is a passive collector of electrons over most of its surface. In a vacuum interrupter with its usual contact gap of 6mm to 15mm, there will be little or no excessive heating of the anode contact at this level of current. Once the arc extinguishes, the old anode now becomes the new cathode during the recovery phase. Experience has shown that in all practical vacuum interrupter designs it is almost impossible NOT to interrupt a current of 2000A (rms.) or less, in most ac circuits.
The above statement, at first sight, seems to be rather a bold one. Let us examine some of the extensive experimental evidence that supports it. One powerful tool that has been in use since the early days of vacuum interrupter development has been to study the recovery rate of the contact gap’s dielectric strength after the extinction of the vacuum arc. At this current level the experiments usually involve a variation of the “free recovery” method that was first developed to investigate the recovery of arcs in air [1]. A vacuum arc is established between the opening contacts. The current can be either ac or dc. Once the contacts have reached a given separation and the vacuum arc has been established for a given length of time, the current is forced to zero using an auxiliary circuit in a time of 0.5ps to 3ps [2, 3]. At a given delay following the arc extinction, the dielectric strength of the recovering contact gap is investigated by applying a step function voltage pulse. The magnitude of this pulse can be varied for a given delay until a dielectric breakdown of the contact gap is obtained. By varying the delay time of the reapplied voltage, a free recovery reignition voltage characteristic as a function of time after current zero can be obtained. Of course, the value of the breakdown voltage for a given time delay can have some considerable variation. The 50% value, or mean value, is quite often used.
A variation of this technique is to apply a fast ramp voltage pulse (say 250kV/ps) after a given time delay and then monitor the voltage at which the dielectric breakdown of the contact gap occurs
[4]. Figure 4.4 shows typical free recovery curves for contact gaps of a few' millimeters after the forced reduction to zero from arc currents of about 200A. Here it can be seen that (a) the recovery rate does depend upon the contact material and (b) the contact gap recovers to its full strength very rapidly even though the di/dt at arc extinction is in excess of ЮОА/ps. From Equation (4.2), this di/ dt is about tw'o orders of magnitude greater than would be expected from the peak value of a 2 kA (rms.) ac current: a current at which a fully diffuse vacuum arc would occur.
In a series of very instructive experiments Lins [57] has measured the metal vapor density in the contact gap after a natural ac current zero for arcing currents in the range 500A (rms.) to 2000A (rms.). Figure 4.5 shows the Cu vapor density before and after current zero of a 500A (rms.) vacuum arc for both metal vapor evaporated from the cathode and also from the molten copper particles exiting the contact gap at 300ms^{1}. It can be seen that at current zero the total Cu density in midcontact gap from the cathode spots and the molten particles is about 10^{17} atoms.m^{3}. Relating these data to Table 4.1 it can be seen that this corresponds to a pressure of less than about 10^{2} Pa even if the contact surfaces are at 2150K. At this pressure the electron mean free path is significantly greater than the contact gap. So, clearly any breakdown of the gap has to be considered a “vacuum breakdown” event and will not have been caused by a Townsend avalanche resulting from electron ionization of the residual metal vapor in the contact gap. In further work using CuCr contacts and currents of lkA (rms.) and 2kA (rms.), Lins has developed Figure 4.6(a) and (b). Note here the data begin 40ps after the current zero. Even so, the residual neutral metal vapor density is still too low
FIGURE 4.4 The free recovery of the vacuum contact gap after a current of about 200A has been ramped to zero in less than 2ps for a diffuse vacuum arc; the CuCr data from [4], the other contact materials [2].
FIGURE 4.5 The Cu vapor density for a 50 Hz, 500A (rms) diffuse, vacuum arc at the center of a 14mm
contact gap before and after current zero: [O] experimental data, [] calculated assuming the cathode has
a temperature of 2000 К and an effective erosion rate of 3pg/C, [] calculated vapor contribution from
Cu molten droplets in flight, which begins with a temperature of 2000 K, a diameter of 10 pm and a velocity of 300ms^{1} [6].
to allow a Townsend avalanche to take place. Even when a 200A, vacuum arc between Cu contacts is forced to zero within lps the residual Cu vapor left between the contacts is still only about
1.2 x 10^{18} atoms.nr^{3}, lps after the current zero, i.e., a pressure of less than 10^{_l} Pa; see Figure 4.7. Although the density of this residual metal vapor is about one hundred times greater than that for a 2kA (rms) arc, Figure 4.6(b), and it decays very slowly (e.g., to about 5 x 10^{17} atoms.nr^{3} after 200ps) the residual metal vapor will have little or no effect on the free recovery of the 2mm contact gap.
The internal pressure inside a practical vacuum interrupter using CuCr contacts after current interruption is affected by two gas adsorption processes. First, the getter placed inside the vacuum interrupter during manufacture: see Section 3.5.1 in this volume. Second, the gettering effect of the Cr deposited on the vacuum interrupter’s internal shield. Weuffel et al. [8] show that for dc arcs with durations (t_{arc}) ranging from 3ms to 960ms there is always a decrease in the internal pressure of the vacuum interrupter: See, for example, Figure 4.8. For t_{arc} > 30ms the adsorption of the residual gas by the metal deposited from the CuCr contacts plays a significant role in the reduction of the internal pressure of the vacuum interrupter. Also, for 460A dc arcs with a duration greater than 30ms the internal pressure is almost independent of the vacuum interrupter’s internal pressure before the arcing occurred.
Another powerful way of examining the recovery of the contact gap’s dielectric strength is to use the WeilDobke “syntheticcircuit” approach [9]. A typical circuit that has been used by Wilkening et al. [10] is shown in Figure 4.9. It consists of a lowvoltage, highcurrent, ac pow'er supply on the
TABLE 4.1
An Estimation of the Gas Density, Electron Mean Free Path and Metal Vapor Mean Free Path as a Function of Pressure and Temperature
Pressure 
Pressure x 10mm contact 8^{a}P 
Temperature, К 

mbar 
Pa 
mbar.mm 
Pa.m 
300 
1350 

Number density, n.m"' 
Approx, electron mean free path,mm 
Approx, metal atom mean free path,mm 
Number density, n.m"' 
Approx, electron mean free path,mm 
Approx, metal atom mean free path,mm 

10' 
10^{s} 
IO^{1} 
10' 
2.4 x IO^{25} 
4 x IO^{4} 
3 x 10' 
5.3 x IO^{24} 
1.8 x 10 ' 
1.4 x IO^{4} 
IO^{2} 
10* 
10’ 
IO^{2} 
2.4 x IO^{24} 
4 x 10' 
3 x IO'^{4} 
5.3 x 10^{2}' 
1.8 x 10^{2} 
1.4 x 10 ’ 
I0 
10' 
I0^{2} 
10 
2.4 x 10» 
4 x IO'^{2} 
3 x 10' 
5.3 x IO^{22} 
1.8 x 10' 
1.4 x 10^{2} 
I 
10^{2} 
10 
1 
2.4 x IO^{22} 
0.4 
3 x IO’^{2} 
5.3 x 10^{21} 
1.8 
0.14 
10' 
10 
1 
10' 
2.4 x 10^{21} 
4 
0.3 
5.3 x 10^{20} 
18 
1.4 
10’^{2} 
1 
10' 
io^{2} 
2.4 x !0^{20} 
40 
3 
5.3 x IO^{19} 
1.8 x 10^{2} 
14 
10’ 
io^{1} 
io^{2} 
io’ 
2.4 x IO^{19} 
4 x IO^{2} 
30 
5.3 x IO^{18} 
1.8 x 10’ 
1.4x IO^{2} 
Pressure 
Pressure x 10mm contact gap 
Temperature, К 

mbar 
Pa 
mbar.mm 
Pa.m 
1700 
2150 

Number density, n.m^{_}' 
Approx, electron mean free path,mm 
Approx, metal atom mean free path,mm 
Number density, n.m^{}' 
Approx, electron mean free path,mm 
Approx, metal atom mean free path,mm 

10' 
10^{5} 
IO^{1} 
10’ 
4.3 x !0^{24} 
2.2 x 10' 
1.7 x IO^{4} 
3.4 x 10^{24} 
2.8 x 10’ 
2.1 x IO^{4} 
Ю^{2} 
IO^{1} 
10' 
10^{2} 
4.3 x 10» 
2.2 x 10^{2} 
1.7 x 10 ' 
3.4 x 10» 
2.8 x 10^{2} 
2.1 x 10’ 
10 
10' 
10^{2} 
10 
4.3 x IO^{22} 
2.2 x IO^{1} 
1.7 x IO^{2} 
3.4 x IO^{22} 
2.8 x IO^{1} 
2.1 x IO^{2} 
1 
IO^{2} 
10 
1 
4.3 x 10^{21} 
2.2 
0.17 
3.4 x 10^{21} 
2.8 
0.21 
io^{1} 
10 
1 
io^{1} 
4.3 x 10^{20} 
22 
1.7 
3.4 x 10^{20} 
28 
2.1 
10^{2} 
1 
10' 
10^{2} 
4.3 x 10^{19} 
2.2 x 10^{2} 
17 
3.4x IO^{19} 
2.8 x 10^{2} 
21 
10' 
10' 
io^{2} 
10’ 
4.3 x 10^{18} 
2.2 x 10' 
1.7 x 10^{2} 
3.4x Ю^{18} 
2.8 x 10’ 
2.1 x 10^{2} 
FIGURE 4.6 The vapor density of Cu and Cr from CuCr contacts after current zero after interrupting 50 Hz, ac currents of 1000A (rms) and 2000A (rms): [O] Cu data, [ Д] Cr data [7].
lefthand side and a highvoltage, low current, high frequency power supply on the righthand side. The main current flow, a half cycle (50Hz) ac wave, begins at time t_{0} from a low voltage circuit. At this time, the high voltage power supply is isolated from this circuit. At time t, the contacts of the vacuum device being tested open and the vacuum arc is initiated. This can then give up to about 9ms of arcing. Longer arcing times allow a larger contact gap at current zero. An isolation switch in
FIGURE 4.7 The Cu vapor density in a 2mm contact gap after the current for a 200A vacuum arc is ramped to zero in less than 2 ps [4].
FIGURE 4.8 An example of the change in the internal pressure inside a vacuum interrupter with Cu Cr(25wt.%) contacts with active and passive getters after interrupting 460A dc current at t = 0 [8]
series with the vacuum contacts opens at time t_{2}. This switch interrupts the main current at time t_{4}. At time tj a high frequency current from the parallel high voltage power supply is injected through the open vacuum contacts. Then the vacuum arc continues operating to the natural current zero of this high frequency current at time t_{5}. During the time interval (t_{5}t_{4}) the isolation switch fully recovers to its design withstand voltage and thus isolates the main current source from the high voltage, high frequency power supply. When the high frequency current passing through the vacuum contacts goes
FIGURE 4.9 The WeilDobke synthetic circuit for evaluating the interruption performance of a vacuum interrupter [10].
to zero, the vacuum arc extinguishes and a characteristic TRV from the high voltage power appears across the open vacuum contacts. The high voltage circuit can be tuned to vary the di/dt at current zero, the RRRV and the peak TRV. Using this approach, the researcher can closely relate the effects of the vacuum arc at different ac current levels and the effects of different contact materials on the dielectric recovery of the contact gap at current zero for realistically shaped TRVs. In the experiments reported by Lindmayer et al. [9], their TRV has a frequency of 25kHz, i.e., the TRV peak is reached in about 20ps, which is about the time that one would expect in a normal, ac, inductive circuit. At currents below 2500A, CuCr contacts will interrupt even this TRV 100% of the time.
Wang et al. [11] use a modified WeilDobke circuit to observe the free recovery of a 12mm contact gap after interrupting a 2.1kA arc current using Cu, CuCr(25wt.%) and CuCr(50wt.%) contacts. In their experiments, a 90kV voltage pulse with a rise time of 150ns (i.e., 460kV/ps) is impressed across the open contact gap at various times after time t_{5}. They show that for 25mm diameter contacts the contact gap recovers between 4ps and 5ps. The metal vapor density decay is similar to that shown on Figure 4.7: i.e., it stays between 1 x 10^{l8}/m^{3} and 3 x 10^{l8}/m^{3} for lOps after time t_{s}.
In order to understand why the diffuse vacuum arc for currents less than 2000A in a typical 50 or 60 Hz ac circuit has no difficulty interrupting that circuit, it is necessary to review the development and operation of this vacuum arc as has already been discussed in Chapter 2 in this volume. As the vacuum interrupter’s contacts begin to open, a molten metal bridge forms. When that bridge ruptures, a highpressure bridge column arc develops in the region previously occupied by the molten metal bridge. This highpressure arc will endure until the evaporation of metal vapor from the arc roots is no longer enough to replace the metal vapor lost to the surrounding vacuum and is also no longer enough to maintain the required arc pressure as the contacts continue to part and the total arc volume increases. When this happens, the diffuse vacuum arc forms with cathode spots (each with a current 50A100A) moving in a retrograde motion over the cathode’s surface with speeds up to 10^{2}m.s^{4}. The cathode spots produce electrons, ions, neutral metal vapor and metal particles. For currents below 2000A, the ions have energies up to about 50eV and have speeds up to 10^{4}m.s~'. The electrons have energies in the range of leV4eV and have speeds of the same order as the ions.
The ions and electrons, for the most part, leave the cathode spot in a cone whose crosssectional angle is about 70°. About 80% of the cathode erosion is in the form of particles, which move away from the cathode spot mostly at an angle of less than 30° from the cathode’s surface. Once the plasma component of the diffuse vacuum arc has left a cathode spot it will have crossed the contact gap (typically 6mm to 15mm) or will have exited the contact region in a time of lps to 2ps. As the ac current decreases towards zero, the number of cathode spots decreases in order to maintain the 50A100A per spot. When a cathode spot ceases to exist, the region of its demise cools extremely rapidly to the temperature of the ambient metal. The thermal time constant of the region close to the spot’s location is on the order of microseconds because the spot’s diameter is so small. Thus, for most practical contact materials, once a cathode spot has extinguished and its temperature drops below 1000°K the continued evaporation of contact material will be negligible; see Table 4.2. As the current proceeds to current zero in an ac circuit, where the rate of change of current is so much slower than the time for plasma dispersal and also very significantly slower than the after effects of cathode spot extinction (e.g., at 2000A rms. di/dt at current zero is about lA/ps), the vacuum arc itself seems to be anticipating the interruption process. To paraphrase Farrall [12]:
a vacuum interrupter will begin to recover from a diffuse vacuum arc while the arc is still burning, just
after the sinusoidal peak. ...At extinction, the contact gap only “remembers" a small period of arcing
just preceding extinction.
At current zero, as I have already discussed, Lins’s and Wang et al.’s experiments show that the residual metal vapor has a pressure of 10^{_1} Pa to 10^{2} Pa. At these pressures the electron mean free path is much greater than the contact gap. Thus, you would not expect the residual metal vapor to play a role in the reestablishment of the vacuum arc between the contacts. For a diffuse vacuum
TABLE 4.2
Vapor Pressure of Contact Metals at Increasing Temperatures
Metal 
Temperature, К 

300K 
1000K 
1360K 
1500K 
1750K 
2000K 
2150K 
2300K 

Vapor Pressure, Pa 
Vapor Pressure, Pa 
Vapor Pressure, Pa 
Vapor Pressure, Pa 
Vapor Pressure, Pa 
Vapor Pressure, Pa 
Vapor Pressure, Pa 
Vapor Pressure, Pa 

Bi 
< io^{s} 
5.3 
1.3 x 10’ 
6.7 x IO^{3} 
> IO^{5} 
> IO^{5} 
> IO^{5} 
> IO^{5} 
Cu 
< io^{s} 
2 x 10^{ft} 
5.3 x 10^{2} 
9.3 x 10' 
27 
4 x IO^{2} 
1.3 x 10^{3} 
4x IO^{3} 
Cr 
< io^{s} 
1.2 x 10^{8} 
2 x 10' 
6.7 x 10^{2} 
5.3 
133 
6.7 x 10^{2} 
2.7 x IO^{3} 
Ag 
< io^{s} 
5.3 x 10^{4} 
2.7 
40 
1.3 x IO^{3} 
6.7 x 10^{3} 
2.1 x IO^{4} 
10^{s} 
W 
< io^{s} 
< io^{8} 
< io^{8} 
< io^{8} 
< io^{8} 
< io^{8} 
< io^{8} 
2.7 x IO^{7} 
arc the ions carry about 10% of the current and the electrons carry the remainder. Just before current zero both the electrons and the ions are moving away from the last cathode spot as shown in Figure 4.10(a). Just after current zero, the former anode becomes the new cathode and the former cathode becomes the new anode. The ions thus continue their motion towards the new cathode, but the electrons very quickly reverse their direction towards the new anode, see Figure 4.10(b). So, the stage is set to discuss the changes that occur between the contacts after the current zero as the TRV appears across them.
The discussion begins with the work by Johnson et al. [13] who have analyzed what would happen when an identical floating double probe is placed in an ionized plasma and is separated from the source that created the plasma. When a difference in potential is applied between the probes, the floating system rapidly adjusts itself so that the more positive probe will be close to the plasma potential. That is, neither probe becomes greatly positive with respect to the plasma nor are large electron currents drawn to them. Also, the potential difference applied to the probes does not penetrate the main body of the plasma but appears across a space charge sheath adjacent to the negative probe. As the current in a vacuum arc approaches very close to the ac current zero, the intercontact plasma in the vacuum interrupter adjusts itself to maintain quasi charge neutrality. In fact, for
FIGURE 4.10 A schematic showing the motion of the electrons and ions from the last cathode spot before and after current zero and the initial rise of the transient recovery voltage.
currents of less than 2000A (rms.), the vacuum arc chops out just before current zero (see Section 2.3.3 in this volume). All emission of electrons and ions from the last cathode spot ceases. A low impedance plasma now bridges the intercontact gap as the restored voltage appears across the open contacts. The ions, which have considerable inertia, will continue to move toward the new cathode (the former anode). This residual current is termed the postarc current (i_{pac}). The electrons, however, will rapidly decelerate, come to a halt (e.g., if they are moving at 10~'ms^{_1}, it will take a potential of only 10~^{5}V to achieve this) and try to reverse direction. The conditions will thus be created for the ions and electrons to rapidly form a stationary, quasineutral plasma between the contacts. This is similar to the double floating probe discussed above. During the brief adjustment period (typically about lps) there will be little voltage drop across the positive space charge sheath that appears at the new cathode. Soon, however, the potential drop from the TRV will appear across the sheath. There is practically zero voltage drop across the rest of the plasma, i.e., the plasma has the same potential as the new anode, see Figure 4.11. Figure 4.12 shows an example of a typical postarc current pulse (i_{pac}) that would be observed in a vacuum interrupter with a floating shield [14]. The current crosses zero at time t = 0, then reaches a peak postarc current, i_{pac}, in the opposite polarity. The di_{pa}/dt during this initial stage is similar to that of the vacuum arc just before the current zero. This results from the ion current flowing to the new cathode, which has an initial velocity of up to l()^{J}ms^{_l} before the neutral plasma is established and the positive space charge sheath is established. During the initial period t_{d} there is a delay in the appearance of the TRV across the contacts. After an initial sharp decrease in i_{pac} there follows a flatter region until t_{k}. Interestingly, up to this time the shield potential has remained zero, i.e., the shield has been connected to the anode by way of the conducting plasma. At the time t_{k}, when there is a distinct drop in i_{pac}, the shield voltage begins to follow the capacitive distribution.
FIGURE 4.11 A schematic illustrating the sheath growth model during the recovery phase of intercontact region after the interruption of a diffuse vacuum arc and the initial rise of the transient recovery voltage [14].
FIGURE 4.12 A schematic showing the current zero region and the development of the post arc current during the rise of the transient recovery voltage [14].
The postarc current results from the ions in the sheath region falling freely towards the new cathode. In the plasma, the electrons carry this postarc current as they withdraw towards the new anode. As they do this, they increase the thickness of the space charge sheath at the cathode. For the most part, the circuit’s TRV appears only across the sheath. It is only when the sheath reaches the new anode that TRV is impressed across the full contact gap. Also, at this time the postarc current ceases. At current levels less than 2000 A, discussed in this section, the contact gap can reignite only after current zero if a cathode spot forms on the new' cathode (the former anode). As this former anode is quite passive during the arcing phase, its surface will be at a comparatively low temperature during the postarc current phase. Thus, the only way to initiate a cathode spot at the new cathode is to develop a field at the new cathode high enough to develop the vacuum breakdown process discussed in Chapter 1 in this volume. The free recovery experiments (see Figures 4.44.7) show that the dU/dt after current zero has to be extremely rapid so that the field across the sheath U_{R}(t)/s(t) is high enough to initiate a reignition of this contact gap. In most practical ac circuits, the dU/dt is usually about an order of magnitude (i.e., 10 to 20 times) low'er than those given in Figures 4.44.7. The work by Wilkening et al. [10] and others has shown that low current vacuum arcs do not reignite even in the face of an extremely fast TRV across an open contact gap after a current zero.
There has been considerable modeling of the postcurrent zero period and of the postarc current phenomena [1425]. This modeling has been useful in providing a predictive quality to the above qualitative explanation of the physical changes between the electrical contacts after current zero. In order to outline the general analysis, I w ill use the model of the post current zero period presented by Fenski and Lindmayer [14] and their cow'orkers at the Technische Universitat Braunschweig. Just after current zero, the contact gap is filled with a neutral plasma. A space charge sheath of thickness s(t) develops in front of the new cathode (the old anode). The sheath’s edge is driven towards the new anode (the old cathode) by the rising TRV. As a first approximation this sheath contains only ions and the whole voltage from the TRV is impressed across it. The remaining neutral plasma has the potential of the new anode. Within the sheath the ions fall freely toward the cathode, giving rise to the postarc current. In the neutral plasma, electrons carry this postarc current toward the anode as the sheath is withdrawn. Thus, the postarc current would be expected to cease when the sheath reaches the new anode. The current density y, is related to the motion of the sheath’s edge by:
where y, is the current density carried by the ions, n, is the ion density at the boundary between the sheath and the neutral plasma, Z is the mean ion charge (typically 1.8), e is the electron charge (1.602 x 10~^{19} A.s), v_{if} is the ion velocity at the sheath’s edge, and s is the sheath’s thickness, see Figure 4.11. Immediately after current zero it can be assumed that the ions are moving towards the new cathode with a directed velocity v_{0} of about 10^{4}ms^{_l}. For typical contact gaps of about 10mm, these ions will reach the new cathode in about lps. During this time, the plasma is undergoing the adjustment period, at the end of which the initial ion velocity v_{f} would be expected to be close to zero. This can be represented in the model by:
with t_{d}=l /us. The ion density «, is being reduced with time as a result of recombination (n, = n_{0}exp{t/x_{R}}), but the flowing postarc current can also be increasing if there is a high enough density of metal vapor in the contact gap. This, however, is unlikely for ac arc currents of 2000A or less. This can be represented by:
where i_{pac} is the postarc current and V_{p} is the plasma volume. The value of the ion decay time constant r_{R} depends upon the arc current and the internal volume of the vacuum interrupter. For freely recovering arcs r_{R} is 0.5ps for a 2mm contact gap and lOps for a 10mm contact gap [26]. For a practical vacuum interrupter switching an ac current, values between 20ps and lOOps are more reasonable. Thus, for low currents where the i_{pac} is over in a few microseconds the ion density can be assumed to be constant. The second term on the righthand side of Equation (4.5) results from the ions moving toward the cathode from the edge of the sheath. It is also possible that there will be an increase in the charge density as a result of the production of new charge carriers during the rise of the TRV. One process that has been postulated [27] is the secondary emission of electrons as a result of positive ions with energy e Z U(t) (where U(t) is the voltage drop across the sheath, i.e., the TRV, U_{R}(t)) impinging on the cathode. If y(U) is the yield at voltage U(t) then the total current i_{pm}. is given by:
y(U) has been shown to rise linearly with voltage so that at 20kV, y(U)=3. Using Equations (4.3) and (4.6):
where, to a first approximation, A is the area of the contacts. A limitation to the actual postarc current that flows between the recovering contacts is the space charge limit imposed by the Child Langmuir equation, i.e.:
where m, is the ion mass and e„ is the permittivity in vacuum, 8.85 x 10"^{12} A.s.V'.nr^{1}. This equation is not quite as accurate as that used by Andrews and Varley [16], but it is valid over most of the postarc sheath [17] and thus is perfectly acceptable as a first approximation. For a simulation to proceed, a firstorder differential equation is required that describes the electrical circuit and another that describes the relationship between the TRV and the current at the interrupter’s terminals.
At current zero s = 0. As discussed above, for a short time after current zero when the polarity across the contacts reverses, the voltage across them stays nominally at zero. The total current i_{pat}. during this time follows the precurrent zero di/dt, because of the inertia of the ions moving with a velocity v_{h} so:
This equation implies that the simulation should begin at time t_{d} in Figure 4.12 where the sheath formation begins, once the electrons have reversed and the electron current is zero, i.e., for (t > t_{d})
where v_{ie} is the velocity of the ions moving from the edge of the space charge sheath to the new cathode. This can be the starting point for the postarc current model. As stated above, the postarc current should cease when the sheath reaches the new anode, i.e., ds/dt = 0 and at t_{k} in Figure 4.12. There always is, however a residual current that remains after t_{k}, i.e., after the sheath reaches the new anode. The cause of this is most probably the slow' decay of the charges remaining in the intercontact gap. Van Lanen et al.’s analytical model gives the postarc current during the rise of the TRV as [22]:
Where e electron charge, n, ion density, v_{B} = (kT/nij), m, ion mass, T_{e} electron temperature, к Boltzmann’s constant, A_{eff} effective cathode area and C_{sh} sheath capacitance. Mo et al. [24] and Takahashi et al. [21] use the PICMCC software (Plasma in Cell with Monte Carlo Collisions software [28]) to model the postarc current. The PICMCC code is a simulation model for plasmas that includes kinetic effects. The simulation requires a lot of computer memory and its run time can be rather long.
For ac currents up to 2kA (rms.), the value of the postarc current is very small. Its duration is a function of the final contact gap at current zero. As the density of metal vapor in the intercontact gap at current zero is so low the usual concept of the postarc current “heating” the residual gas (see, for example, in gas circuit breakers as reported by Frost et al. [29]), does not apply in low current vacuum arcs. Also, at a fully open contact gap, it is unlikely that a new cathode spot will form from almost all normal TRVs, so the possibility of a reignition after current zero is extremely remote. It is now possible to give an explanation for the rapid free recovery data shown in Figure 4.4. In this
Figure, the CuCr contact with a 2mm contact gap recovers to its full withstand strength of 90kV in about 4ps after the low current (200A) vacuum arc has been ramped down to zero in just under 1.5ps; see Lins [4]. The ion density in this contact gap as measured by Lins [19] is about 3 x 10^{l7}m~^{3 }just before and just after the fastcurrent ramp to zero. The ion density drops to below 10^{l5}nr^{3} in about 3.5ps: the decay constant, r_{R}, being 0.5ps. Lins speculates that the decay of the ion density and the recovery of the contact gap during this time must, in some way, be related. The ions crossing the contact gap during the free recovery period would impact the new cathode and liberate electrons that could contribute to the gap’s breakdown during the 4ps to 5ps period. As the fastest ions from the last cathode spot would cross the contact gap in lps, the question remains why does it take 4ps to 5ps for the contact gap to fully recover? Wang et al. [11] conclude that during the initial breakdown period (Ops to 4ps or 5ps) the probability of breakdown drops with the decay of the ion current during the free recovery period. The collision of the ions with the residual metal vapor slows their decay so that it only drops below a critical density after 4ps or 5ps: see Figure 4.13. There seems to be a direct relationship between the decay of the ion density and the free recovery of the contact gap for a 2kA vacuum arc. In order for the vacuum arc to reignite after the current is ramped to zero the cathode has to develop a cathode spot capable of sustaining the resulting vacuum arc. Because the neutral gas density (Figure 4.7) and the ion density are so low, there will be no interaction of any electrons that may be liberated from the new cathode. Thus, any reignition must develop from the vacuum breakdown process described in Chapter 1 in this volume enhanced by the ion bombardment at the new cathode.
Up to about lps after the current zero, no voltage appears across the contacts. This is the plasma adjustment period. After this period the sheath is formed at the cathode: all the TRV voltage U_{R}(t) appears across the sheath. The sheath moves toward the new anode with a velocity given by Equation (4.7). If U_{R}(t)/s(t) has a value greater than 4.5 x 10'Vnr^{1}, then a high enough electron current could be liberated by field emission from the new cathode, which, in turn, could initiate the vacuum breakdown process. Equation (4.7) shows that ds/dt is inversely proportional to the ion density n,. The ds/ dt is slower just after current zero than 3ps later when the n, is 0.01 of its value at current zero. Thus dU_{R}(t)/dt has to be extremely rapid to create an electric field at the cathode that is high enough to result in the vacuum breakdown of the contact gap after the 200A arc, even though, in this case, the
di/dt of the current just before the current zero is higher than lOOA.ps^{1}. In Figure 4.4 the average dU_{R}(t)/dt for CuCr contacts is about 21kV.ps^{1}, which means that the average ds(t)/dt has to be less than 0.5mm.ps^{4}.