APPENDIX A
FORCE EXERTED ON A DRIVE COIL DUE TO NEIGHBORING DRIVE COILS

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 4_m /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 30^o 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:

Coil no.1
Coil no.2
Coil no.3
Coil no.4
Coil no.5
Coil no.6
Coil no.7

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 3_m 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 2_m; ENTER; key in the dM/dx for a distance of 3_m (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/i_ai_b.)

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 -180^o t 180^o.

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.

001	*LBLA	034	RCL5	067	x
002	DEG	035	9	068	RCL3
003	ST03	036	0	069	x
004	R	037	-	072	ABS
005	STO2	038	X<0?	071	x
006	R	039	GTOB	072	ABS
007	STO1	040	GSBc	073	RTN
008	CF3	041	STO	076	SIN
009	1	042	GSBb	075	RCL5
010	STO4	043	ST-0	076	SIN
11	PSE	044	GSBa	077	X2
12	PSE	045	ST-0	078	RCL2
13	F3?	046	GTO3	079	x
14	X2	047	*LBLB	080	RCL4
15	ST04	048	GSBc	081	x
16	CF3	049	ENT	082	ABS
17	*LBL1	050	+	083	RTN
18	CLX	051	CHS	084	*LBLc
19	PSE	052	STO	085	RCL5
20	F3?	053	GSBb	086	SIN
21	GTO2	054	ST-0	087	RCL5
22	GTO1	055	*LBL3	088	COS
23	*LBL2	056	RCL0	089	x
24	X<0?	057	F2?	090	RCL1
25	SF2	058	CHS	091	x
26	ABS	059	PRTX	092	RCL4
27	STO5	060	PRTX	093	x
28	1	061	GTO1	094	ABS
29	8	062	*LBLa	095	RTN
30	0	063	RCL5	096	*LBLe
31	-	064	SIN	097	CLX
32	X>0?	065	RCL5	098	1/X
33	GTOe	066	COS	099	RTN
				100	R/S

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