We can recast the equation of motion.

Equation 4.1, in dimensionless form by dividing through by the breakaway torque T_B.  We obtain

The quantity J_aθ_g/T_B has units of (time)², so we define a characteristic time t_a for a magnetic breaker as

This “time” is seen to be a function of the inertia of the armature, the spring constant of the armature restraining spring, the amount of pre-load we place on the restraining spring, and the angular width of the armature air gap.  Each different magnetic breaker will have its own particular value of t_a.  But in terms of t_a, all magnetic breakers of similar design will have the same dynamic behavior.

If we assume the idealized case of no latch mechanism load torque (i.e. T_latch = 0), we will not need to consider impact effects when the armature slams into the latch mechanism.  This case will also be the minimum time response, or fastest possible response, for a magnetic structure.  The normalized equation of motion can then be written as

We can easily solve this differential equation numerically for any given set of values for the frame reluctance fraction m, normalized coil current i/I_th, and gap angle – spring pre-load angle ratio θ_g/θ_o.  Example solutions for m = 0.1, θ_g/θ_o  = 1.0, and different values of step input i/I_th are given in Figure 4.9.

The latch mechanism trips the contact separation mechanism when the armature angle reaches the latch threshold value θ_th.  The time required to trip the contact separation mechanism, measured from the overcurrent inception, is the detection time of the breaker t_d.  We plot the detection time response for the example solutions given in Figure 4.9 as a function of input normalized overload current in Figure 4.10.

In this case we have assumed that the threshold armature angle is 80% of the armature gap angle.  As can be seen, the normalized detection time is a monotonically decreasing function of the normalized input current.

In the case of no armature saturation, at high values of input current (in the example of Figure 4.10 for i>5I_th), the detection time becomes linearly dependent on the ratio I_th/i.  This linear behavior at high input currents can be deduced directly from the normalized equation of motion.  At high input currents the armature angular acceleration is approximately proportional to (i/I_th)² only.  Thus a simple double integration gives

From which we see that

for a given threshold angle θ_th.

If armature saturation occurs, the time response of the armature will be slowed slightly.  The detection times for a saturated case would be slightly higher than non-saturated values, such as those given by the solid line in Figure 4.10.Saturated values would tend to level off at a minimum detection time since the drive torque would also saturate at a maximum value, given by (4.9) using the saturated gap flux value.  A typical saturation case is shown by the dashed line response in Figure 4.10.