We show in Figure 5.15a the complete steady-state DC solution for a flat arc heat flux potential with arc current I = 10i_o.

 In Figure 5.15A we also show a simple one term approximation solution  

Where the mid-plasma space magnitude S_m is given by the exact value (See Equation 5.32)

As is evident, this one term solution is an excellent approximation for the exact solution, given by (5.13) and (5.21) for this relatively high level of current drive.

Now consider the low current drive exact, and approximate solution for I = 0.1i_o given in Figure 5.15b.  Here we see that the one term approximate solution overestimates the plasma region width by a factor of

But when considered as a fraction of the total interwall space this overestimate is only 14%.

Figure 5.16 illustrates a summary comparison of the exact and approximate solutions for the plasma region halfwidth as a function of plasma steady-state current.  The worst case overestimate is seen to occur near the i/i_o = 0.1 drive point.  And here, as stated above, it is only a 14% overestimate.  The approximate solution for the heat flux potential given by Equations 5.33 and 5.34 is thus seen to be a “safe” or conservative estimate for the spatial variation of the heat flux potential of the actual flat arc.  We will use this one term approximation as the basis for a simplified dynamic arc model.

The one term transient solution for a step driven change in the heat-flux potential is taken from (5.30).

where the initial state magnitude S_mo and the final state magnitude S_mf are determined from the initial and final steady-state currents and (5.34).  The single arc thermal time constant τ is given by

The transient value of the plasma region halfwidth x_i is the x_i root of

which, in closed form, is

The transient current electric field relationship is given by

where

Performance Equations 5.35 through 5.40 are ideally suited for numerical analysis since they are based on stepwise change which, in an incremental sense, is the basis for all numerical solution techniques.  For example, assume that a flat arc is driven by a sinusoidal current source.  We approximate the continuous drive current by a stepped, discrete, staircase current, such as that shown in Figure 5.17. 

At the end of each step we solve for the resultant new electric field E(t) and plasma region halfwidth x_i using (5.35), (5.38), (5.39) and (5.40).  We then reassign these values as initial values for the next time step and repeat the solution process.  Typical results are shown in Figures 5.18 through 5.20 for ratios of arc thermal time constant τ to drive frequency period T, τ/T = 1.0, 0.1 and 0.01.

In these current driven results we note particularly the collapse of the arc, as indicated by the halfwidth x_i, as the current goes through a zero crossing.  For a given drive frequency period T, we see that the arc collapse tends to follow the current zero crossing very closely for “fast” time constant arcs (i.e. τ<<T).  But for “slow” time constant arcs (i.e. when τ is comparable to T), the arc collapse is incomplete and lags behind the current zero crossing.

In terms of practical circuit breaker design, we desire fast time constant arcs.  Fast arcs will fully collapse at a current zero crossing and thus not be subject to thermal reignition when the arc contact voltage redevelops.  From the arc thermal time constant expression (5.36), we see that in order to obtain a fast arc we need only confine the arc in narrow arc channels.  The arc time constant is seen to vary as the square of the channel width.  To confirm this theoretical prediction, Frind [5.11] conducted careful measurements of the time constants of flat arcs using the apparatus shown in Figure 5.21. 

The results of Frind’s measurements are shown in Figure 5.22. 

The dotted straight line in Figure 5.22 indicates adependency rather than the theoretical square relationship.  However in view of the inability to hold the channel walls at a constant temperature and the rough approximations that were made in deriving (5.36), the agreement can be considered to be quite satisfactory.  At very small channel widths, near 20 mils, the measured time constant were extremely fast, approximately 0.1 μsec.  This encouraging result alone, not withstanding our theory, suggests the advisability of arc confinement in cool arc channels.