The Interaction of the Vacuum Arc and a Transverse Magnetic Field
The Diffuse Vacuum Arc and a Transverse Magnetic Field
In the literature on the vacuum arc, the term “Transverse Magnetic Field” really refers to an “Transverse Magnetic Flux.” In order to prevent confusion, I will generally adopt the commonly used expression “Transverse Magnetic Field” instead of the more correct “Transverse Magnetic Flux” (interestingly, the abbreviation TMF can be used interchangeably for both expressions). The observation that the cathode spots from a low-current diffuse vacuum arc move away from each other in a retrograde (i.e., an anti-Amperian) motion has a long history, This has led to experimental studies of a single cathode spot subjected to an external transverse magnetic field “Br” impressed across it: see, for example, Fang  and Persky et al. . Wang et al.  show that a single cathode spot with a current of 40A and a contact gap of 4mm moves in a retrograde motion faster for BT = 180mT than for BT = 20mT. see Figure 2.51. Figure 2.52 shows that the speed “v” of a single cathode spot on both Cu and Cu-Cr (25wt%) cathodes increases as a function of B,i
where P* is a constant and B, ranges from 20mT to 200mT. Nemchinsky  modeled the retrograde motion of a cathode spot in the presence of a transverse magnetic field. He explains the
FIGURE 2.51 Single, 40A, cathode spot’s retrograde motion on a Cu cathode with a contact gap of 4mm, for three values of external transverse magnetic fields 
FIGURE 2.52 Retrograde velocity of single 40A cathode spot on Cu and Cu-Cr (25 wt%) cathodes as a function of the transverse magnetic field .
retrograde motion in terms of the voltage drop fluctuations, theHall field created by surface positive charges at the retrograde side of the cathode spot and the probability of the spot jumping. His model gives a cathode spot velocity for a spot current of 30A and a transverse field of 80mT of about 6-7 ms-'. This velocity is similar to that given by Wang et al.  for a cathode spot on Cu with a 40A current shown in Figure 2.52. Shi et al.  present a similar relationship for the B, range 20mT to 1250mT. They also develop a random-walk model of a single cathode’s spot motion in a Br. The relationship between the external BT and the ignition probability of a new cathode spot is analyzed. They assume that the ignition probability is proportional to the magnetic pressure around the former cathode spot. Their model uses Beilis’ hypothesis  that the ignition probability of the new cathode spot is highest at the retrograde side of the cathode crater where the magnetic pressure is the highest. Chaly et al.  also show that cathode spots move in a retrograde manner when subjected to a transverse magnetic field BT.
Alferov et al. [135, 136] show that a dc current can be interrupted by applying an external Br across a diffuse vacuum arc. They open Cu-Cr (50wt%) contacts in dc circuits ranging from 100A to 200A with Br values from lOOmT to 200mT. Initially as the contacts begin to open a low voltage diffuse vacuum arc with multiple cathode spots is formed. As the contacts continue to open the contact gap increases and a stage is reached when the interaction of the BT with the cathode spots begins to take effect. The arc voltage becomes wildly unstable with oscillations that can reach greater than 1000V. Figure 2.53 shows an example of the highly variable voltage across a single cathode spot subjected to an external Br . When the B, is applied across the diffuse vacuum arc in a dc circuit, the high voltages cause a decrease in the dc current until the unstable vacuum arc extinguishes and the dc circuit is interrupted [135, 136]. Figure 2.54 illustrates this effect for an 80A vacuum arc in a 500V circuit 
Consider the open contacts shown in Figure 2.55(a) . The cathode spots are spread over the contact surface. They emit the ions and electrons that make up the neutral plasma carrying a current density in the intercontact space. If a uniform transverse magnetic field, BT, is suddenly impressed across the contacts as is shown in Figure 2.55(a), then the plasma will be dominated by the Hall Field EH given by:
where ne is the electron density and e the electron charge. EH is the transverse field required to move electrons across the magnetic lines of force. Its direction is shown in Figure 2.55a. At a distance (see Figure 2.55b) from a cathode spot emitting an electron current ie and an ion current
FIGURE 2.53 The voltage excursions shown by a single 40A cathode spot subjected to an external, transverse magnetic field, BT .
FIGURE 2.54 The effect of an external magnetic field on cathode spots showing the interruption of a dc current .
ij we have j = -ie/(2кг2) and n = /,/(2nr2e), where v, is the mean ion velocity. The mean Hall field is therefore:
FIGURE 2.55 (a) Contact configuration, showing the current and Hall field in a plasma unperturbed by the
transverse magnetic field: (b) coordinates used in the model and the calculation; (c) sketch of the plasma configuration as a pulsed transverse magnetic field is applied across the plasma .
Now we know that the ratio ic/i, is constant for a given material and is in fact roughly equal to 10 for most materials [73, 83]. The average Hall field is, therefore, independent of current. The Hall force on the ions eE„ is greater than the Lorentz force ~(ev,Br) by the large factor ie//, and is opposite in sense. For that reason that force dominates the motion of the ions, and bends the plasma in the “forward” or Amperian direction.
The Hall field is large, EH = 35 V/cm when Br■ = 0.05T in the case of Cu and is transverse. If the cathode spots were to remain spread over the surface of the cathode, as in Figure 2.55(a), the Hall field would cause a variation in sheath potential across the cathode. It is reasonable to assume that there is only one potential at which a spot can burn stably; therefore, as indicated in Figure 2.55(c), the cathode spots must be aligned near a single line of force. A calculation of the plasma structure that uses the coordinates of Figure 2.55(b)  shows that most of the current is restricted to a thin shell on the retrograde side of the plasma, as indicated in Figure 2.55(c). The calculated plasma structures are showrn in Figure 2.56. The calculated dense region near the plasma boundary results from the reflection of ions at the boundary, and all reflected ions lying between the boundary and the internal reflection envelope. Experimental photographs of this plasma show a similar structure, but it is markedly more diffuse .
Once the ions head in a direction parallel to the anode surface, in order to drive the circuit current into the anode, the electrons will have to flow across the gap ‘d’ between the neutral plasma and the anode. The voltage across the contact gap will increase to a value given by the Child’s Law for space charge limited current :
where К = 5690V/(A)2/3. This effect has been used, as I have discussed above, by Alferov et al. to interrupt dc currents up to 400A in 200V circuits. It has also been used to successfully develop a
FIGURE 2.56 A sketch of the calculated plasma structure .
high-voltage, dc, circuit breaker that can interrupt currents up to 15kA for circuit voltages up to 80kV ; see Section 6.11.
The Columnar Vacuum Arc and a Transverse Magnetic Field
I will discuss the practical application of the transverse magnetic field (TMF) to control the columnar vacuum arc in Chapter 3 in this volume (Section 3.3.3), when I discuss practical vacuum interrupter contact structures. Here, let us consider a columnar vacuum arc that has formed between the two rails shown in Figure 2.57. If an external TMF of strength, Вр is impressed on the columnar vacuum arc then a force FT in the direction shown will be imposed upon the column arc of length La, carrying a current i, which is given by:
In a practical contact design, the current flow in the rails is shown in Figure 2.57. Here the current flowing in the rails supplies the TMF, BT The actual value of BT in the region of the columnar arc is not obvious. At a position ‘P’ a good distance behind the arc column, the current in the rails has a
FIGURE 2.57 A columnar vacuum arc between conducting rails.
more or less uniform distribution and B, = BT‘ can easily be calculated. On the other side of the arc column there is no current flow and J5r is negligible. Michal  investigated this problem analytically and experimentally and has shown that BT in the region of the arc column is about half that at the position “P” for La = 3mm. In other experiments on arcs between runners in air  and an analysis of a columnar arc in vacuum  shows that Equation (2.36) can be modified for the force FT on the arc column to be:
where n* « 0.5. In the configuration shown in Figure 2.57, we would expect the arc column to move in the direction shown. Shmelev’s 1-D model  of the columnar vacuum arc gives the ratio of the cathode temperature Tc and the anode temperature TA as Tc / TA я 1. The columnar plasma in the contact gap determines the direction of its travel in the Amperian direction. This direction is now one that would be expected from a metal conductor placed between and in contact with the rails; i.e., in the Amperian direction. Thus, the columnar vacuum arc behaves differently from the diffuse vacuum arc where the individual cathode spots move in an anti-Amperian motion or retrograde motion. It has been found experimentally that when an arc forms between opening contacts it does not immediately move under the influence of this BT The arc roots initially dwell at the location where they are first established. It is only after a certain dwell time (or perhaps after a minimum value of arc length is reached) that the arc column begins to move [141, 144, 145]. Once the arc does begin to move, there will be opposing forces on the column arc from the arc roots attached to the cathode and to the anode. The arc roots cannot move instantaneously and thus will limit the speed of the arc motion. Delachaux et al.  and Shmelev et al.  have developed models of the high-current columnar vacuum arc with a B, impressed across it. They use a 2-D MHD model together with a radiation transfer approximation. The attachment at the cathode is assumed to be a solid uniform area. Both models also assume that the arc roots at both contacts have a temperature in excess of 3000K. This is somewhat higher than experimental data that gives anode temperature of a columnar vacuum arc to be between 2000K and 3000K [148, 149].
A schematic of the arc column motion is shown in Figure 2.58 . It illustrates a possible sequence for its travel. The plasma jet from the anode advances beyond the main arc column and heats the cathode area in the Amperian direction. Once this surface reaches a high enough temperature (perhaps greater than 2200K), there will be a copious emission of metal vapor that will allow the cathode attachment to jump to its new position. The anode attachment will then follow a
FIGURE 2.58 The motion of the columnar vacuum arc subjected to a transverse magnetic field .
few microseconds later. In vacuum there is no drag force on the arc that results from the arc motion through an ambient atmosphere. There are, however, drag forces resulting from momentum lost by neutral atoms escaping from the arc column as it moves  and the reluctance of the arc roots on the contacts to move to a new location. When the driving force from the magnetic field is greater than the drag forces the arc will begin to move.