12th WSEAS Int. Conf. on APPLIED MATHEMATICS, Cairo, Egypt, December 29-31,

Transcription

1 th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 5 Magnetostatic Field calculations associated with thick Solenoids in the Pesence of Ion using a Powe Seies expansion and the Complete Elliptic Integals. VASOS PAVLIKA Haow School of Compute Science Univesity of Westminste, School of Compute Science, Haow, Middlesex, UK Abstact:- The effect of ion on the unifomity of the field poduced by an axisymmetic thick solenoid is consideed. Using a powe seies expansion of the vecto potential in the adial and axial coodinates the potential is found and fom this the magnetic induction B is deived. The solution to the vecto potential and field components is also achieved using the complete Elliptic Integals of Legende of the fist and second kind espectively with numeical esults using both methods of solution computed. Key Wods:- Time independent field, Powe Seies expansion and Elliptic Integals of the fist and second kind.. Intoduction. In this pape magnetostatic field calculations associated with an axisymmetic conducto of ectangula coss section situated equidistant fom two semi-infinite egions of ion of finite pemeability ae computed. The magnetostatic field associated with ion-fee axisymmetic systems has been consideed by Boom and Livingstone [], Gaett [] and many othes. Caldwell [3], Caldwell and Zisseman [] and [5] have caied out wok which takes account of the effects of the pesence of ion on such systems. The main advantages of intoducing ion ae: i. Highe fields ae povided fo the same cuent, poducing substantial powe savings ove conventional conductos. ii. The field unifomity is impoved even fo supeconducting solenoids by placing the ion in a suitable position. The geomety consideed is shown in figue, a tooidal conducto V of ectangula coss section having inne adius A, oute adius B and length L-, is located equidistant between two semi-infinite egions of ion of finite pemeability a distance L apat, the axis of the conducto being pependicula to the ion boundaies. The egion V between the conducto and the ion is assumed insulating. Cylindical coodinates (,φ,) ae used whee and ae nomalied in tems of L.. Poblem Fomulation Pio to Caldwell [3] the pesence of ion in axisymmetic systems had been lagely ignoed see ion a V V Figue. A tooidal conducto V of ectangula coss section located midway between two semi infinite egions of ion of finite pemeability. The egion V is assumed to be insulating. Loney [6] and Gaett [] et al. In cylindical coodinates Maxwell s equations give: B = in V Ce in V ' φ whee e φ is a unit vecto in the diection of inceasing φ and C is a constant with. B = in V and V () Equation () suggests the intoduction of a potential A such that B = A, axial symmety Aφ ( Aφ ) implies B = ; Bφ = ; B = By Maxwell s equation: b ion

2 th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 59 in V B= ( A) = Ceφ in V ' thus e eφ e in V = ϕ Ceφ in V ' Aφ ( Aφ ) inv Aφ = Ceφ inv ' whee = ϕ with bounday conditions fo A φ A φ = on = ; A φ as Aφ = on = and =. Axial symmety gives. Using the integal epesentation of the ϕ vecto potential this gives () A() = dv', hence fo finite μ, ' A (, ) = φ V ' μ b π xcos ϑdxdϑd' K / π a n= {( ' n) x xcos ϑ) () whee K μ =, known as the image facto. μ Noting that Aφ (, ) is an odd function in and an even function in then A φ can be expanded as a powe seies about the axis giving: (3) A K I m φ(, ) = μ m() n= m= whee equation () gives b I() = [[ wlog e x α ]] a = with w= --n and α = x w. Substituting expession (3) into equation () gives I () n m m m K ( mm ) Im() = n= m= m= equating coefficients of I m () Im () mm ( ) Im( ) =, m=,,,... m m () I () so that Im() = m mm!( )! m m () I () m and A φ(, ) = μ K m n= m= mm!( )! To elate this to the wok of Gaett [] let b a( x, w) = wlog x α I = [[ a( x, w)] ] A ( ) = b whee a ( x, w) = [[ wlog x α ] a] = () b and Am = [[ am( x, w)] a] = (5) m m ()()! ma m so that A φ(, ) = μ K m n= m= mm!( )! m m ()( m)! A m = μ K n= m= (m )! whee (m-)!=.3.5 (m-), and (m)!=..6 (m), m a with Am = (6) m m so fo the field components m m ()( m)! A m () B (, ) = μ K and n= m= ( m)! m m () (m )! A m () B (, ) = μ K n= m= (m )! 3 5 Hence A φ(, ) = μ K A A3 A5... n= 6 3 B (, ) = μ K A A3 A5... n= and e a = B (, ) =μ K A A A6... n= 6 The fist five tems will be quoted, the emainde can be obtained fom the ecuence elations equations (), (5) and (6). So that x A = [[ log ( x ( w x ) )] ] / ( w x ) x xw b A3 = [[ ] ] / 3/ a = ( w x ) ( w x ) x 3xw A = [[ 3/ 5/ ( w x ) ( w x ) xw b ]] 3/ a = ( w x ) / b e a =

3 th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 6 3xw 6xw A = [[ ( w x ) ( w x ) 5 5/ 3/ 3 5xw x ( w x ) ( w x ) 7/ 3/ 3xw b ]] 5/ a = ( w x ) and 9x 9xw A = [[ ( w x ) ( w x ) 6 5/ 3/ 3 5xw xw ( w x ) ( w x ) 5xw 7/ 7/ ]] b a = 3. The solution to the Magnetic Vecto Potential using the Complete Elliptic Integals. In ode to compae and validate the esults of the pevious section an independent solution fo the magnetic vecto potential and hence field components B (,) and B (,) must be deived. Using equation () if the integation with espect to ϑ is done fist, the complete Elliptic Integals of Legende of the fist and second kind espectively ae obtained. Defining π x cosϑ = dϑ / ( β γ cos ϑ) whee β = w x and γ=x which can be witten as: x π cosϑ = dϑ / β ( k cos ϑ) γ whee k = β, so that β π k cosϑ = d / ϑ ( kcos ϑ) β π / β π dϑ = ( kcos ) d / ϑ ϑ ( kcos ϑ) with slight manipulation this can be witten as β π/ π/ du = ( ) ( sin ) / k δ udu π/ π/ / ( k) (δ sin u) k ϑ π whee δ = and = u k so that I = β ϑ /( ( k) E( δ) K( δ) ) ( β γ) whee K(δ) and E(δ) ae the complete Elliptic integals of Legende of the fist and second kind espectively. Povided <δ< these integals may be expessed as a seies which is unifomly convegent and thus may be integated tem by tem. So consideing this inequality with: x k x k = and δ = = w x k w ( x ) x hence fo convegence < < w ( x ) i.e. x > which is tue x, >. Similaly the second inequality gives x < w ( x ) o ( x ) < w, which is again tue xw,,. So that the seies is unifomly convegent. Hence using π K( δ) = δ δ δ O( δ )...6 and 3 π.3 δ.3.5 δ E( δ) = δ O( δ ) gives μ b n n β Aφ (, ) = K { k / a n ( β γ) n= δ.3 ( k ) δ ( k ) δ ( k 6)...} dxd' Consideing the Highe Ode, n tems of δ. Consideing the ode δ tem which will be denoted by I, say whee b n β k I = dxd' a n / ( β γ) u I = dudw / ( w u ) whee x=u and w=-, so that μ A K u u w w u / φ(, ) = [[( )log e( ( ) ) π n= / / b n ww ( u) w log( eu ( w u) )] a ] ' = n O( δ ) Evaluating the highe ode tems as shown in Pavlika [], it can be shown that: /

4 th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 6 μ A K w u w u / φ(, ) = [[ log( e ( ) ) n= 3 w ( w u) / (7u 7w u w) 3/ 6 uw( w u ) w(59 u 9 w) (u 6uw uw w) 3/ / 3 u ( w u ) uw( w u) 6 w(3u w) 3/ - 3 u ( w u ) (u uw 3u w u w w) 3/ ] b ] n a ' = n ( w u ) Oδ ( ) (6) o μ Aφ (, ) = K [[ α3,(, uw) α3, (, uw) n= 3 α3,3(, u w) α3,(, u w) b n α3,5(, ) α3,6(, ) α3,7 (, )] a ] = ' n u w u w u w O( δ ) whee the α i,, i=3, =,3, 7 ae defined by expession (6) 5. Calculating the Radial and Axial Field Components. Since B = A, using cylindical coodinates this gives μ B (, ) = K [[ α3,(, uw) 3 α3,(, uw) n= 3 α ( u, w) 5 α ( u, w) 3,3 3, 5 6 b n α3,5 α3,6 α3,7 = ' 6 ( uw, ) 7 ( uw, ) ( uw, )] a ] n Oδ ( ) (7) and diffeentiating with espect to gives B μ K u w u / (, ) = [[ {log( e ( ) ) n= w } u / / 3/ ( w u ) ( u ( w u ) ) ( w u ) 3 u(9u uw w) (59u 9 w ) 5/ 5/ w( w u ) 3( w u ) 5 3 u(u uw w) 5/ ww ( u) 6 ( w u ) 5/ w(5u uw w) 5/ ] b ] n a ' = n - Oδ ( ) ( w u ) () Results fo Aφ (,), B(, ) and B(, ) using expessions (6), (7) and () with a=.9, b=., =.5 and μ = wee found to be in good ageement with the solution using the Powe seies expansion as shown in tables,, 3, and Conclusions The two methods of solution wee found to be in good ageement. The summations wee pefomed fom - to with a change only in the fouth decimal place occuing when the numbe of tems in the summation was doubled. The effect of the pemeability of the ion is shown in figues, 3, and 5. The two methods descibed can be easily computeised and povide a quick and flexible method fo calculating and thus demonstating the effects of the pemeability of ion μ, on the field components. It is clea that the accuacy of the methods and the egion of applicability can be extended by taking moe tems in both the Powe Seies and in the seies obtained using the Complete Elliptic Integals of the st and nd kind espectively. The effects of the ion in boosting and impoving the field homogeneity ae clealy evident. BH,L Figue. The vaiation of B (,) with and fo two semi-infinite egions of ion of unit pemeability. :=.3, *:=., :=.

5 th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 6 BH,L Figue 3. The vaiation of B (,) with and fo two semi-infinite egions of ion of infinite pemeability. :=.,*:=., :=.3 BH,L [] Gaett, M.W., Axially symmetic systems fo geneating and measuing magnetic fields. J. Appl. Phys.,, 9 (95). [3] Caldwell. J., Magnetostatic field calculations associated with supeconducting coils in the pesence of magnetic mateial, IEEE, Tansactions on Magnetics, Vol. MAG-,, 397 (9). [] Caldwell, J and Zisseman A., Magnetostatic field calculations in the pesence of ion using a Geen s Function appoach. J.Appl. Phys.D 5,, (93a). [5] Caldwell, J and Zisseman A., A Fouie Seies appoach to magnetostatic field calculations involving magnetic mateial accepted fo publication in J.Appl. Phys (93b). [6] Loney, S.T., The Design of Compound Solenoids to Poduce Highly Homogeneous Magnetic Fields. J.Inst. Maths Applics (966), -5. [7] Pavlika, V., Vecto Field Methods and the Hydodynamic Design of Annula Ducts, Ph.D thesis, Univesity of Noth London, Chapte II, 995. [] Pavlika, V., Vecto Field Methods and the Hydodynamic Design of Annula Ducts, Ph.D thesis, Univesity of Noth London, Chapte III, Figue. The vaiation of B (,) with and fo two semi-infinite egions of ion of unit pemeability. :=., *:=., :=.3 BH,L Figue 5. The vaiation of B (,) with and fo two semi-infinite egions of ion of infinite pemeability. :=., *:=., :=.3 7. Refeences [] Boom, R.W., and Livingstone. R.S., Poc. IRE, 7 (96).. Tables μ= 3 μ= μ= μ= Table. Values of A φ (,) using the Powe Seies Expansion accuate O( ). μ= 3 μ= μ= μ=.. 5.5E E E E E E E E

6 th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, Table. Values of B (,) using the Powe Seies Expansion accuate O( 5 ). μ= 3 μ= μ= μ= Table 3. Values of B (,) using the Powe Seies Expansion accuate O( ). μ= 3 μ= μ= μ= Table. Values of A φ (,) using the Elliptic Integals of the st and nd kind, accuate O(δ ). μ= 3 μ= μ= μ=.. 5.3E E E E E E E E Table 5. Values of B (,) using the Elliptic Integals of the st and nd kind, accuate O(δ ).