Forces and torques between two square current loops

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1 Int. J. of Applied Electromagnetics and Mechanics 10 (1999) IOS Press Forces and torques between two square current loops Z.J. Yang 1, J.R. Hull Argonne National Laboratory, Argonne, IL 60439, USA J.A. Locwood 2 and T.D. Rossing Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA Received 27 May 1997 Revised 24 March 1998 Abstract. We derive closed-form expressions for the magnetic forces and torques between two coaxial square current loops as a function of their relative tilt angle. These expressions involve simple numerical integrations. An analytic expression for the torsion constant at zero tilt angle is derived for loop filaments and for loops of finite axial extent. Keywords: magnetic bearing, magnetic force. 1. Introduction Current loops and coils are widely used in electromagnetic applications, such as motors, actuators, coil guns, magnetic levitation and positioning, etc. [1. Current loops, in the guise of Amperian currents, can also be used in calculations that involve permanent magnets. Several papers have provided analytic or closed-form expressions for the forces in systems composed of coils or magnets with fixed and uniform polarization, not necessarily coaxial, but with zero tilt angle. Aoun and Yonnet presented an analytic expression for the forces exerted between two cuboidal magnets [2. This expression has been used to calculate torques in magnetic couplings [3, but at zero tilt angle. Calculation of forces and torques for circular geometries typically requires numerical integration [4 6. In this paper, we study a system that consists of two square coaxial current loops. First, we derive the formula for the interactive forces between two square filamentary current loops, the centers of which are aligned but tilted with respect to each other. The resultant closed-form expression is given in terms of elementary functions and integrals that are easily treated numerically. Next, based on the above results, we present a formula for the torque acting on the tilted loop. By taing the derivative with respect to tilt angle of this formula, we obtain an analytic expression for the torsion constant at zero tilt angle. This torsion constant is useful when calculating the stability of magnetic levitations, and an analytic expression for this term greatly decreases the time needed to evaluate the stability of a particular configuration. Finally, we obtain the torsion constants for loops of different axial extent. 1 Corresponding author. Present address: Lucent Technologies, 2000 N. Naperville Rd., Naperville, IL 60566, USA. Fax: ; zjyang@lucent.com. 2 Present address: Commonwealth Research Corporation, Chicago, IL 60623, USA /99/$ IOS Press. All rights reserved

2 64 Z.J. Yang et al. / Forces and torques between two square current loops 2. Two finite straight wires First, we derive the magnetic induction from a straight current element of length 2a, as shown in Fig. 1. According to classical theory, the magnetic induction can be expressed as [7 B 1 = µ 0I 1 a a (r r ) r r 3, (1) where = dz, r = xi + yj + z, r = z,andi, j, and are the unit vectors along the three axes of the Cartesian coordinates. Direct integration yields B 1x (x, y, z)= µ 0I 1 B 1y (x, y, z)= µ 0I 1 B 1z (x, y, z)=0. y x 2 + y 2 x x 2 + y 2 [ z a x 2 + y 2 +(z a) 2 [ z a x 2 + y 2 +(z a) 2 z + a x 2 + y 2 +(z + a) 2, (2) z + a x 2 + y 2 +(z + a) 2, (3) Next, we derive the force between two finite straight current wires as illustrated in Fig. 2. From Ampère s law, the force acting on the second straight wire with length 2L can be calculated as Let F = I 2 B 1. L (4) x = ξ + x, y = η + y, z = ζ + z, (5) Fig. 1. Schematic diagram for deriving magnetic induction around a finite straight current wire with length 2a and current I 1.

3 Z.J. Yang et al. / Forces and torques between two square current loops 65 Fig. 2. Schematic diagram for deriving forces from a first wire acting on a second finite-length straight current wire with length 2L and current I 2. where (ξ,η,ζ) are the coordinates for the center of the second straight current wire. Thus, we have = dx i + dy j + dz =(i sin θ cos φ + j sin θ sin φ + cos θ). (6) Explicitly, the three components of the force can be expressed in the form F x = µ 0I 1 I 2 cos θ ξ + l sin θ cos φ L (ξ + l sin θ cos φ) 2 +(η + l sin θ sin φ) 2 [ G(ξ,η,ζ,θ,φ,a, l) G(ξ,η,ζ,θ,φ,a,l), (7) F y = µ 0I 1 I 2 cos θ L η + l sin θ sin φ (ξ + l sin θ cos φ) 2 +(η + l sin θ sin φ) 2 [ G(ξ,η,ζ,θ,φ,a, l) G(ξ,η,ζ,θ,φ,a,l), (8) F z = µ 0I 1 I 2 sin θ L ξ cos φ + η sin φ + l sin θ (ξ + l sin θ cos φ) 2 +(η + l sin θ sin φ) 2 [ G(ξ,η,ζ,θ,φ,a, l) G(ξ,η,ζ,θ,φ,a,l), (9) where the function G(ξ,η,ζ,θ,φ,a,l) is defined as G(ξ,η,ζ,θ,φ,a,l)= ζ + l cos θ + a (ξ + l sin θ cos φ) 2 +(η + l sin θ sin φ) 2 +(ζ + l cos θ + a) 2. (10) Now, let us consider four special cases: θ = 0, π/2, and φ = 0, π/2.

4 66 Z.J. Yang et al. / Forces and torques between two square current loops Case 1. θ = 0 When θ = 0, the function G is simplified to G(ξ,η,ζ,0,φ,a,l)= ζ + l + a ξ 2 + η 2 +(ζ + l + a) 2. (11) Substituting Eq. (11) into Eqs (7) (9), we obtain the forces as F x (θ = 0)= µ 0I 1 I 2 + F y (θ = 0)= µ 0I 1 I 2 + F z (θ = 0)=0. Case 2. θ = π/2 ξ ξ 2 + η 2 [ ξ 2 + η 2 +(ζ + L a) 2 ξ 2 + η 2 +(ζ L a) 2 ξ 2 + η 2 +(ζ L + a) 2 ξ 2 + η 2 +(ζ + L + a) 2, (12) [ η ξ 2 + η 2 ξ 2 + η 2 +(ζ + L a) 2 ξ 2 + η 2 +(ζ L a) 2 ξ 2 + η 2 +(ζ L + a) 2 When θ = π/2, the function G is simplified to G(ξ,η,ζ,π/2,φ,a,l)= ξ 2 + η 2 +(ζ + L + a) 2, (13) ζ + a (ξ + l cos φ) 2 +(η + l cos φ) 2 +(ζ + a) 2. (14) Substituting Eq. (14) into Eqs (7) (9), we obtain the forces as F x (θ = π/2)=0, F y (θ = π/2)=0, F z (θ = π/2)= µ 0I 1 I 2 8π { [sgn(ζ + a) sgn(ζ a) ln L 2 + ξ 2 + η 2 + 2L(ξ cos φ + η sin φ) L 2 + ξ 2 + η 2 2L(ξ cos φ + η sin φ) + sgn(ζ + a) ln G 1(ξ,η,ζ,φ,a,L) G 1 (ξ,η,ζ,φ,a,l) + sgn(ζ a) ln G 1(ξ,η,ζ,φ,a, L) G 1 (ξ,η,ζ,φ,a, L) }, (15) where sgn(x) denotes the sign-function, and the function G 1 (ξ,η,ζ,φ,a,l) is defined as G 1 (ξ,η,ζ,φ,a,l)=2(ζ + a) 2 + L 2 + ξ 2 + η 2 + 2L(ξ cos φ + η sin φ) + 2 ζ + a (ζ + a) 2 + L 2 + ξ 2 + η 2 + 2L(ξ cos φ + η sin φ). (16)

5 Z.J. Yang et al. / Forces and torques between two square current loops 67 Case 3. φ = 0 When φ = 0, the function G is simplified to G(ξ,η,ζ,θ,0,a,l)= ζ + l cos θ + a (ξ + l sin θ) 2 + η 2 +(ζ + l cos θ + a) 2. (17) Substituting Eq. (17) into Eqs (7) (9), we find that the integrals cannot be analytically realized. Case 4. φ = π/2 When φ = π/2, the function G is simplified to G(ξ,η,ζ,θ,π/2,a,l)= ζ + l cos θ + a ξ 2 +(η + l sin θ) 2 +(ζ + l cos θ + a) 2. (18) As in Case 3, the integrals in Eqs (7)(9) cannot be analytically realized by this simplification. 3. Two square loops 3.1. Forces Using the results obtained in the previous section, we derive the formulas for the forces between the two square current loops with lengths of 2a and 2L, as illustrated in Fig. 3. Each square loop consists of four pieces of straight wire, labeled a 1, a 2, a 3,anda 4 for the active loop, and b 1, b 2, b 3, and b 4 for the passive loop, as illustrated in the figure. For convenience, we list the all components for all sixteen pairs explicitly as follows: Fig. 3. Schematic diagram for deriving forces from a first (active) square current loop, 2a 2a, acting on a second (passive) square current loop, 2L 2L.

6 68 Z.J. Yang et al. / Forces and torques between two square current loops 1. a 1 b 1 F a1 b 1,x = µ 0I 1 I 2 L cos α a 2π (h + L sin α) 2 +(Lcos α a) 2 [ (h + L sin α) 2 +(Lcos α a) 2 +(La) 2 (h + L sin α) 2 +(Lcos α a) 2 +(L + a) 2, (19) F a1 b 1,y = 0, F a1 b 1,z = µ 0I 1 I 2 h + L sin α 2π (h + L sin α) 2 +(Lcos α a) 2 [ (h + L sin α) 2 +(Lcos α a) 2 +(La) 2 (h + L sin α) 2 +(Lcos α a) 2 +(L + a) 2. (20) 2. a 1 b 2 F a1 b 2,x = 0, F a1 b 2,y = µ 0I 1 I 2 8π F a1 b 2,z = 0. { [1 sgn(l a) ln L 2 + h 2 + a 2 + 2L(h sin α a cos α) L 2 + h 2 + a 2 2L(h sin α a cos α) + ln G 1(h, a, L, π/2 α, a, L) G 1 (h, a, L, π/2 α, a, L) + sgn(l a) ln G 1(h, a, L, π/2 α, a, L) G 1 (h, a, L, π/2 α, a, L) }, (21) 3. a 1 b 3 F a1 b 3,x = µ 0I 1 I 2 L cos α + a 2π (h L sin α) 2 +(Lcos α + a) 2 [ (h L sin α) 2 +(Lcos α + a) 2 +(La) 2 F a1 b 3,y = 0, (h L sin α) 2 +(Lcos α + a) 2 +(L + a) 2, (22)

7 Z.J. Yang et al. / Forces and torques between two square current loops 69 F a1 b 3,z = µ 0I 1 I 2 h L sin α 2π (h L sin α) 2 +(Lcos α + a) 2 [ (h L sin α) 2 +(Lcos α + a) 2 +(La) 2 (h L sin α) 2 +(Lcos α + a) 2 +(L + a) 2. (23) 4. a 1 b 4 F a1 b 4,x = 0, F a1 b 4,y = F a1 b 2,y, F a1 b 4,z = 0. (24) 5. a 2 b 1 F a2 b 1,x = µ { 0I 1 I 2 [sgn(a L 2 +(h + L sin α) 2 + a 2 2aL L cos α)+1 ln 8π L 2 +(h + L sin α) 2 + a 2 + 2aL + sgn(a L cos α) ln G 1(h + L sin α, a, L cos α, π/2,a,l) G 1 (h + L sin α, a, L cos α, π/2,a,l) ln G } 1(h + L sin α, a, L cos α, π/2, a, L), (25) G 1 (h + L sin α, a, L cos α, π/2, a, L) F a2 b 1,y = 0, F a2 b 1,z = a 2 b 2 F a2 b 2,x = µ 0I 1 I 2 F a2 b 2,y = µ 0I 1 I 2 F a2 b 2,z = µ 0I 1 I 2 sin α h + l sin α L (h + l sin α) 2 +(La) 2 [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l), (26) cos α L L a (h + l sin α) 2 +(L a) 2 [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l), (27) cos α L h + l sin α (h + l sin α) 2 +(L a) 2 [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l). (28)

8 70 Z.J. Yang et al. / Forces and torques between two square current loops 7. a 2 b 3 F a2 b 3,x = µ { 0I 1 I 2 [1 L 2 +(hlsin α) 2 + a 2 2aL sgn(l cos α a) ln 8π L 2 +(hlsin α) 2 + a 2 + 2aL F a2 b 3,y = 0, F a2 b 3,z = a 2 b 4 F a2 b 4,x = µ 0I 1 I 2 F a2 b 4,y = µ 0I 1 I 2 F a2 b 4,z = µ 0I 1 I 2 9. a 3 b 1 + ln G 1(h L sin α, a, L cos α, π/2,a,l) G 1 (h L sin α, a, L cos α, π/2,a,l) + sgn(l cos α a) ln G 1(h L sin α, a, L cos α, π/2, a, L) G 1 (h L sin α, a, L cos α, π/2, a, L) sin α h + l sin α L (h + l sin α) 2 +(L + a) 2 }, (29) [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l), (30) cos α L L + a (h + l sin α) 2 +(L + a) 2 [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l), (31) cos α L h + l sin α (h + l sin α) 2 +(L + a) 2 [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l). (32) F a3 b 1,x = µ 0I 1 I 2 L cos α + a 2π (h + L sin α) 2 +(Lcos α + a) 2 [ (h + L sin α) 2 +(Lcos α + a) 2 +(La) 2 (h + L sin α) 2 +(Lcos α + a) 2 +(L + a) 2, (33) F a3 b 1,y = 0, F a3 b 1,z = µ 0I 1 I 2 h + L sin α 2π (h + L sin α) 2 +(Lcos α + a) 2 [ (h + L sin α) 2 +(Lcos α + a) 2 +(La) 2 (h + L sin α) 2 +(Lcos α + a) 2 +(L + a) 2. (34)

9 Z.J. Yang et al. / Forces and torques between two square current loops a 3 b 2 F a3 b 2,x = 0, F a3 b 2,y = µ 0I 1 I 2 8π F a3 b 2,z = a 3 b 3 { [1 sgn(l a) ln L 2 + h 2 + a 2 + 2L(h sin α a cos α) L 2 + h 2 + a 2 2L(h sin α a cos α) + ln G 1(h, a, L, π/2 α, a, L) G 1 (h, a, L, π/2 α, a, L) + sgn(l a) ln G 1(h, a, L, π/2 α, a, L) G 1 (h, a, L, π/2 α, a, L) F a3 b 3,x = µ 0I 1 I 2 a L cos α 2π (h L sin α) 2 +(alcos α) 2 [ (h L sin α) 2 +(alcos α) 2 +(La) 2 }, (35) (h L sin α) 2 +(alcos α) 2 +(L + a) 2, (36) F a3 b 3,y = 0, F a3 b 3,z = µ 0I 1 I 2 h L sin α 2π (h L sin α) 2 +(alcos α) 2 [ (h L sin α) 2 +(alcos α) 2 +(La) 2 (h L sin α) 2 +(alcos α) 2 +(L + a) 2. (37) 12. a 3 b 4 F a3 b 4,x = 0, F a3 b 4,y = F a3 b 2,y, F a3 b 4,z = 0. (38) 13. a 4 b 1 F a4 b 1,x = F a2 b 1,x, F a4 b 1,y = 0, F a4 b 1,z = 0. (39) 14. a 4 b 2 F a4 b 2,x = F a2 b 4,x, F a4 b 2,y = F a2 b 4,y, F a4 b 2,z = F a2 b 4,z. (40) (41) (42)

10 72 Z.J. Yang et al. / Forces and torques between two square current loops 15. a 4 b 3 F a4 b 3,x = F a2 b 3,x, F a4 b 3,y = 0, F a4 b 3,z = 0. (43) 16. a 4 b 4 F a4 b 4,x = F a2 b 2,x, F a4 b 4,y = F a2 b 2,y, F a4 b 4,z = F a2 b 2,z. (44) (45) (46) Summing all of the terms gives the following equations for the total forces that are from the first square current loop acting on the second square current loop: F total,x = F a1 b 1,x + F a1 b 3,x + F a3 b 1,x + F a3 b 3,x + 2F a2 b 2,x + 2F a2 b 4,x, (47) F total,y = 0, F total,z = F a1 b 1,z + F a1 b 3,z + F a3 b 1,z + F a3 b 3,z + 2F a2 b 2,z + 2F a2 b 4,z. (48) In particular, when α = 0wehave F total,x (α = 0)=0, [ 3µ 0 I 1 I 2 h F total,z (α = 0)= π[h 2 +(La) 2 h 2 + 2(L 2 + a 2 ) h 2 + 2(L a) 2 [ 3µ 0 I 1 I 2 h π[h 2 +(L + a) 2 h 2 + 2(L 2 + a 2 ) h 2 + 2(L + a) 2, (49) 0 for I 1 I 2 > 0, and it is easy to show that F total,z α = 0. α= Torque The torque acting on the second loop along the y-axis is T y = L [ F a1 b 3,z + F a3 b 3,z F a1 b 1,z F a3 b 1,z cos α + L [ F a1 b 1,x + F a3 b 1,x F a1 b 3,x F a3 b 3,x sin α + µ [ 0I 1 I L 0 2 l(h + l sin α) 2π 0 L (h + l sin α) 2 +(La) 2 [ G(h, L a, 0,α,0, a, l) G(h, L a, 0,α,0,a,l) µ 0I 1 I 2 2π [ 0 0 L l(h + l sin α) (h + l sin α) 2 +(L + a) 2 [ G(h, L + a, 0,α,0, a, l) G(h, L + a, 0,α,0,a,l), (50)

11 Z.J. Yang et al. / Forces and torques between two square current loops 73 where the function G is defined by Eq. (10). It is easy to show that µ 0 I 1 I 2 h T y (α = 0)= π[h 2 +(La) 2 [ l a l h 2 +(La) 2 +(l a) l + a 2 h 2 +(La) 2 +(l + a) 2 0 µ 0 I 1 I 2 h π[h 2 +(L + a) 2 [ l a l h 2 +(L + a) 2 +(l a) l + a 2 h 2 +(L + a) 2 +(l + a) 2 0 = µ 0I 1 I 2 h 2π { ln [ h 2 +(La) 2 + a 2 a[ h 2 +(L + a) 2 + a 2 + a [ h 2 +(La) 2 + a 2 + a[ h 2 +(L + a) 2 + a 2 a } + ln [ h 2 + 2(L 2 + a 2 )+L + a[ h 2 + 2(L 2 + a 2 )+La [ h 2 + 2(L a) 2 + L a[ h 2 + 2(L + a) 2 + L + a [ µ 0 I 1 I 2 h + 2π[h 2 +(La) 2 (L + a) h 2 + 2(L a) 2 +(a L) h 2 + 2(L 2 + a 2 ) 2a h 2 +(La) 2 + a 2 [ µ 0 I 1 I 2 h 2π[h 2 +(L + a) 2 (L + a) h 2 + 2(L 2 + a 2 ) +(a L) h 2 + 2(L + a) 2 2a h 2 +(L + a) 2 + a 2. (51) 3.3. Torsion constant at α = 0 For magnetically levitated systems, it is important to derive the torsion constant at α = 0, which is defined as τ = T y / α α=0. τ = µ { 0I 1 I 2 L h 2 [ L 1 π h 2 +(L + a) 2 h 2 + 2(L 2 + a 2 ) 1 h 2 + 2(L + a) 2 h 2 [ L 1 + h 2 +(La) 2 h 2 + 2(L 2 + a 2 ) 1 h 2 + 2(L a) 2 h 2 + 2(L 2 + a 2 [ ) h 2 +(L + a) 2 h (L + a) 2 [ h 2 +(L + a) 2 2h 2 L h 2 +(L + a) 2 + a 2h 2 L h 2 +(L + a) 2 + a h 2 + 2(L 2 + a 2 [ ) h 2 +(La) 2 h (L a) 2 [ h 2 +(La) 2 2h 2 L h 2 +(L a) 2 a 2h 2 L h 2 +(L a) 2 a }.(52)

12 74 Z.J. Yang et al. / Forces and torques between two square current loops Fig. 4. (a) Torsion constant (in units of µ 0I 1I 2/π) as a function of distance h (in units of a) at various L (in units of a), (b) Torsion constant (in units of µ 0I 1I 2/π) as a function of L (in units of a) at various distances h (in units of a). In two particular cases, we can simplify the results as follows: [ 2µ0 I 1 I 2 al 4aL a τ(h = 0) = 2 + L 2 L + a La π (L 2 a 2 ) 2 L 2 a 2, (53) and 2µ 0I 1 I 2 π a 3 (L a) 2 as L a, τ(l = a) = µ 0I 1 I 2 a 2 [ 3h 2 + 4a 2 ( h π (h 2 + 4a 2 ) a 2 ) h 2 + 4a 2 h 2 ( ) 1 + h 2 + 4a 2 h 2 + 4a 1 4a 2 2 h 2 + 8a 2 h 2 h 2 + 4a 2 2µ 0I 1 I 2 π (54) a 3 h 2 as h 0. (55) To visualize how the torsion constant depends on the distance between the two loops and the size of loops, we plot the torsion constant (Eq. (52)) in Fig. 4 as a function of distance for four typical values of L/a and as a function of the size L/a for various ratios of distance to length h/a Torsion constants for strip geometry If the upper square loop has a finite thicness 2b in the z-direction, the torsion constant is obtained by integrating over the thicness of the strip and the equation reads τ = µ { 0I 1 J 2 L (L + a) arcsinh +(La) arcsinh π 2(L + a) 2(L a) 2L arcsinh 2(a 2 + L 2 )

13 +(L + a) arctan Z.J. Yang et al. / Forces and torques between two square current loops 75 +(La) arctan 2 + 2(a + L) (L a) 2 (L + a) (L + a) arctan (L a) 2 + 2(a 2 + L 2 ) (L a) arctan (L a) (L + a) 2 + 2(a 2 + L 2 ) + L[ 2 + 2(a 2 + L 2 ) 2 + 2(a + L) 2 2 +(a + L) 2 } h+b + L[ 2 + 2(a 2 + L 2 ) 2 + 2(L a) 2 2 +(La) 2, (56) =hb where J 2 is the current density (per unit length/thicness) of the upper loop. Furthermore, if the bottom square loop also has a finite thicness 2c in the z-direction, the torsion constant is obtained by integrating over the thicness of the strip. The final analytical expression is τ = µ [ 0J 1 J 2 L (L + a) arcsinh + (L a) arcsinh π 2(L + a) 2(L a) 2L arcsinh 2(a 2 + L 2 ) + 4L 2 + 2(a 2 + L 2 ) (2L + a) 2 + 2(a + L) 2 (2L a) 2 + 2(L a) 2 + (L + a) arctan + (L a) arctan 2 + 2(a + L) (L a) 2 (L + a) (L + a) arctan (L a) 2 + 2(a 2 + L 2 ) (L a) arctan (L a) (L + a) 2 + 2(a 2 + L 2 ) (L + a)(2l + a) ln 2 + 2(a + L) 2 (L + a) (a + L) 2 +(L + a) (L a)(2l a) ln 2 + 2(L a) 2 (L a) (L a) 2 +(La) (L + a)(2l a) + ln 2 + 2(L 2 + a 2 ) (L + a) (a 2 + L 2 )+(L + a) (L a)(2l + a) + ln 2 + 2(L 2 + a 2 ) (L a) h+b+c hbc +, (57) (a 2 + L 2 )+(L a) =h+bc =hb+c where J 1 is the current density (per unit length/thicness) of the bottom loop. Here, h is the centerto-center distance between the two square loops. 4. Discussion and summary Although we have derived the formulas for square loops, the same approach can be used to derive expressions for rectangular loops. The formulas will also be analytical in terms of elementary

14 76 Z.J. Yang et al. / Forces and torques between two square current loops functions, but with more terms because of reduced symmetry. In engineering designs, a circular loop is often more appropriate than a square loop. However, the circular loop will require elliptical integrals, which maes it difficult to see clearly the relationships among the various parameters. Physically, both circular and square loops should give the same results at a reasonable resolution if we do not wor on an extremely precise design. In Section 3, we have shown that the forces, torque, and torsion constants can be analytically expressed in terms of a few elementary functions. The advantages of these analytical expressions are obvious: we can use them easily without the multidimensional integrals that are involved in circular geometry. Physically, we now the equivalence between a permanent magnet and a current loop [8. The magnetic effects from square filamentary (or strip-shaped) current loops are exactly the same as those from a sheet (or bloc) of the permanent magnet. Thus, the results presented in this paper can be applied to a system consisting of two square permanent magnets by replacing I 1 and I 2 (or J 1 and J 2 for a strip) with m 1 and m 2 (or M 1 and M 2 for a bloc). Here, m 1 and m 2 are magnetic moment densities per unit area of the two magnets. (M 1 and M 2 are magnetizations of the two magnets.) In summary, we have derived the formulas for the interactive forces, torque, and torsion constant between two square filamentary current loops, whose centers are aligned and one of which is tilting toward the other one. We also generalize the result of the torsion constant from filamentary geometry to strip geometry. Acnowledgements This wor was partially supported by the US Department of Energy, Energy Efficiency and Renewable Energy, as part of a program to develop electric power technology, under Contract No. W Eng-38. References [1 S.A. Nasar and I. Boldea, Linear Motion Electric Machines, Wiley, New Yor, [2 G. Aoun and J.-P. Yonnet, 3D analytic calculation of the forces exerted between two cuboidal magnets, IEEE Trans. Magn. 20(5) (1984), [3 J.-P. Yonnet, S. Hemmerlin, E. Rulliere and G. Lemarquand, Analytical calculation of permanent magnet couplings, IEEE Trans. Magn. 29(6) (1993), [4 E.P. Furlani, A formula for the levitation force between magnetic diss, IEEE Trans. Magn. 29(6) (1993), [5 E.P. Furlani, Formulas for the force and torque of axial couplings, IEEE Trans. Magn. 29(5) (1993), [6 K.-B. Kim, E. Levi, Z. Zabar and L. Birenbaum, Restoring force between two noncoaxial circular coil, IEEE Trans. Magn. 32(2) (1996), [7 J.D. Jacson, Classical Electrodynamics, 2nd edn, Wiley, New Yor, [8 D. Crai, Magnetism: Principles and Application, Wiley, New Yor, 1995.

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