During the passage of a bucket coil, a sinusoidal current flow described by the equation
will appear in each drive coil. The angular frequency, of the flow will be 2/period, where the period equals 4m /V with m being the spacing between drive coils and V being the bucket velocity. At any moment, there will be four contiguous drive coils having a current flow within them. Every current will be 30o out of phase with that in each of the adjacent drive coils. Figure 2 illustrates the current flows in neighboring coils during the activation of a particular reference drive coil (coil no. 4). The equations describing the current flows with respect to that reference drive coil current are as follows:
The force experienced by any two active coils will be given by the product of the instantaneous current in each coil multiplied by the gradient of their mutual inductance (dM/dx) at the distance of their separation:
This force will be attractive between coils with currents of the same sign, and repulsive between coil currents of opposite sign.
Refer again to figure 1 and consider the forces acting on the reference coil no. 4. During its first quarter-cycle (- t - /2), coil no.4 will interact repulsively with coil no. 1, repulsively with coil no. 2, and attractively with coil no. 3. At exactly t = - /2, a weakly felt coil no. 1(a distance 3m away) will turn off and a much closer coil (coil no. 5 only a distance m away) will be activated. Because the force on coil no. 4 due to coil no. 5 is stronger than was the force from coil no. 1, a discontinuity in the gradient of the force acting on coil no. 4 will exist. During the second quarter-cycle (- t 0), coil no. 4 will feel a repulsion from both coils nos. 2 and 3, and an attraction to coil no. 5. Through considerations of symmetry, the forces exerted on coil no. 4 in the third quarter-cycle are the negative of those exerted during the second quarter-cycle. Similarly, forces felt during the fourth quarter-cycle are the negatives of those experienced during the first quartercycle. Over the complete cycle, then, a drive coil receives no net force from its neighboring drive coils.
The program listed below is designed to calculate the reaction force on a drive coil due to all other active drive coils in its neighborhood as a function of the time from which that coil was turned on. It is written to be run on a Hewlett-Packard HP-67/HP-97 calculator.
It is assumed that bucket velocity may be taken as constant during the passage of the bucket through any four successive drive coils (a distance of only a few centimeters). Hence the angular frequency will be constant.
The program is initialized by the input of three pieces of information: key in the dM/dx between drive coils separated by a distance m; ENTER; key in the mat a distance of 2m; ENTER; key in the dM/dx for a distance of 3m (ref.4). Initiate the program by pressing the button labeled [A]. Program execution will begin. Very quickly, the program will pause for a second and the display will show "1.0." During this pause, key in the maximum drive coil current. (this value will default to 1.0 of no entry is made; in this case, all final answers will actually be F/iaib.)
The program will then loop, displaying a zero for a second and then blurring for a second. At any instant when the machine has paused with a zero showing, key in a value of t and the reaction force of the drive coil at that instant will be calculated. The program accepts values of t expressed in degrees rather than radians. The range of values -180o t 180o.
Once a force has been calculated, the answer, (in Newtons) is displayed for 10 sec. the program then branches to the zero/blur input mode, ready to have the next value of t keyed in.
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