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Analysis of the transient electromagnetic field in a conducting bushing
Finite Elements in Analysis and Design 11 (1992) 265-274 Elsevier
265
FINEL 239
Analysis of the transient electromagnetic field in a conducting bushing A l e k s a n d e r K. G~siorski Department of Electrical Engineering, Polytechnic of Czestochowa, Poland
Received January 1987 Revised February 1992 Abstract. The paper presents a general method for the analysis of the transient electromagnetic field and the
calculation of power losses in axially-symmetrical conductors with known boundary conditions. The method consists of the numerical version of the Bubnov-Galerkin method (finite element method) for discretization of the conducting region and the one-step method for discretization of time. On the basis of algebraic considerations, numerical computations for the selected shape of the bushing used in practice have been performed. Calculations of the relative power losses in the bushing as a function of time have been performed and graphs based on them are presented.
Introduction
Let us consider a solid of revolution (axially-symmetrical relative to the revolution axis z ) m a d e of isotropic n o n - f e r r o m a g n e t i c conducting material with constant magnetic permeability /-~ = ~0 and conductivity y (Fig. 1). T h e solid is placed in the field of a thin c o n d u c t o r which, starting at t = 0, carries an alternating current (with its transitory c o m p o n e n t ) described by the relation: i ( t o , / 3 , t) = I v ~ - [ s i n ( t o t + a ) - s i n
a exp(-/3t)],
t~>0,
(1)
where w is the angular frequency, and a , / 3 are constant values. T h e thin c o n d u c t o r is placed in space in such a way that on the surface S of the solid, the magnetic field intensity lip is constant and i n d e p e n d e n t of the angle 0 of the cylindrical system (r, 0, z) [1]. F o r such an axially-symmetrical solid, both the magnetic field in its interior and the inducting field have only one non-zero c o m p o n e n t in the 0-direction. The equation describing this c o m p o n e n t takes the form [2,3]: OH V2H - ].~'y-~"- ~-- 0,
(2)
where t~2H 1 aH 02H V Z H = 0r 2 + -r - ar- + -az 2
H r2'
Correspondence to: Dr. Eng. Aleksander K. G]siorski, Department of Electrical Engineering, Polytechnic of Czestochowa, 17 Armii Krajowej Avenue, Czestochowa, Poland.
0168-874X/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
266
,t-tl 'lv
Fig. 1. Axially symmetrical solid placed in the field of a straight conductor with current.
with the initial condition: H(r,
t : O) I t~ : 0
for the entire region of the wire
(3)
and the Dirichlet boundary condition:
H(r,t)[s=Hp.,{s
for t >~0,
(4)
where S is the boundary of the region g2 of the conductor.
Method of solving the problem The analytical solution of eqn. (2) with conditions (3) and (4) is limited to solids with simple cross-sectional shapes [2,4-6]. For much more complex cross-sectional shapes it is necessary to find the solution numerically [7]. The finite element method is being more and more widely used for discretization of regions with complex cross-sectional shapes. The numerical method used in this paper is based on the direct Bubnov-Galerkin method. This method has been used in electrical engineering since 1973 [4-6,8]. In order to arrive at an approximate solution of eqn. (2) with the conditions (3) and (4), it is assumed that the magnetic field distribution in the conducting solid takes the form [5,6]: 1,
H = S, H~,,
(5)
1-- o
where the Ct are basis functions independent of time, disappearing on the boundary, H t are unknown values, but H0~b0 = i ( w , [3, t)/(2vrr). Therefore function H always satisfies, as required, the Dirichlet boundary condition (see eqns. (23) and (28)) regardless of the values of coefficients H l, 1 ~
A. Gqsiorski / Transientelectromagneticfield in a conductingbushing
267
eqn. (5). By making use of the Green formula [6,8]. 0u fagrad v grad u dO + fay V2u dO = -Is [v-dS'On
(6)
we obtain the Bubnov-Galerkin equations system [5,6] for the diffusion equation (2) with the boundary conditions (3) and (4):
grad q,, grad ~,~ + -~,~m
~ H~ l=1
dO + ~ ' ~
~,~mdO = 0,
(7)
where /~ is the conductor region. In the system of equations (7) integrals on the boundary do not appear because they vanish, since ~om I s = 0 for 1 ~
( [Ce] + --[d 1 e]-'~
Hj = 0, Hk
ca
(8)
where [9,11,12] H i, Hi, H k are the values of the magnetic field intensity at vertices i, j, k,
1 (Iaiai aja ii [Ce]= ~ aka
aiaj ajaj akaj
aiakI Ibibi
aja k + bib i aeak bkbi
11]
12
2 1
1 ; 2
bjbj bkbj
bib k bkbk
+-r2 -~
1
2 1
; (9)
(10)
T is the area of the triangular finite element with coordinates (r i, Zi) , (rj, Zj), (rk, zk) for the vertices, given by T = 7' det
I!
ri rj
Zi]
zi ;
(11)
Fk Zk 1 re = 3(r i + rj + rk);
(12)
ai=rk-r~,
(13)
b i = z i - z k.
The remaining coefficients a j, bi, ak, b,, are determined by cyclic changes of the indices i, j, k. Summing the element equations over all finite elements g2e in the region /2, a global equation is formed for relation (7):
-d[Dl~){tt}
[c] + 1
d
=0.
(14)
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
268
i"! d
r
HJ,__+~,_
H(t)
/
;
J
,
l
i:
t+At
±
Fig. 2. P l o t s o f H(t) a n d d H ( t ) / d t v e r s u s t i m e o v e r t h e t i m e i n t e r v a l t to t + A t .
The matrix equation (14) should be valid for every instant of time, including times t and t + At. For these times, it takes the respective forms: l
d
[ C ] { H } , + ~ [ D l d - 7 ( H }, = 0, 1
15)
d
[ C ] { H } , + ~ , + - - [ O ] - d - ; { H } , + . , , = 0. ¢0
16)
Let us assume that in a small time interval A t, the nonlinear continuous function f = d H(t)/d t can be linearly interpolated and approximated by a straight line linking the point Pl(t)= f(t) and P2(t + At)=f(t + At). These points are placed at the begining and end of the time interval (Fig. 2). The straight line can be represented by d
~ H ( t) =at + b,
(17)
where a and b are constant coefficients. From eqn. (17)
f(m'+~'dH= f'~'(at + b) dr.
(18)
{tt}t
After integration, we arrive at {HI t+At
-{U}t=½a[(At+t
) 2 __
t z] + b ( t + A t - t ) .
(19)
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
269
From eqn. (17) d d -d--t-{H}t+at + -~f{H}t=a[(At+t+t) + 2 b ] .
(20)
On comparing the last two expressions, we can write d d 2 ~-7 {H},+a, + ~-7{H}, = ~-7({HIt+a, - {H},).
(21)
After equating the expressions (15) and (16) and using the relation (21), and performing some algebraic transformations, we obtain ( l~__A_7[D] +
½[C]){H}t+at =
( l~_A_7[D] _
½[C]){H}t.
(22)
The procedure adopted above corresponds to a more general analysis provided in [13] and it is called the Crank-Nicholson scheme. In order to find the value of the v e c t o r {H}t+At at the next time step, it is necessary to know its preceding value {H},. For the start value (t = 0) of the vector {H}0 we can simply take the zero value. In the course of the calculation process, the matrices [C] and [D] dependent on the geometry and parameters of the conductor material do not change, which makes it possible to calculate them only once. This reduces the time of calculation. The numerical values of the boundary condition (4), which change in the course of the numerical process and are different for every time instant considered, can be represented on the basis of the flux law [1] and eqn. (1) in the form: 1
lip. t Is = -~-R[sin(oot+a)
-sin a exp(-flt)],
peS,
(23)
where R is the radial distance of boundary point p (see Fig. 1). The other symbols are the same as for eqn. (1). The boundary conditions at every time step will be incorporated into the currently solved system of equations (22). The matrix equation (22) appears to be easy to solve numerically. The numerical procedure, however, is complicated by the necessity of considering the time-varying values of the boundary condition. The procedure for every time step is as follows: The vector {H}t+a, is filled with the boundary values corresponding to the time step t + At. The left-hand side of expression (22) is calculated with the known values of {H}t determined in the previous step, which--after multiplying--is defined as {Fi}t+at. - The right-hand side of expression (22) is calculated and set equal to the vector {Fr}' with {H}t determined in the previous step. The terms of the vector {F}t+at on the right-hand side are determined by subtracting the vector {Fi}t+at from {Fr}t, and then substituting the terms corresponding to the known values of boundary conditions, given in terms of the vector {H}t+a,. - For the vector {H}t+at only those terms placed on the boundary have known values. For every boundary term, it is necessary to modify the left-hand side matrix (1/to At)[D] + ½[C]. To do that, we put zeroes in both the row and column relating to the considered boundary term of the vector {H}t+at and put a value of one in the place of their intersection. Finally the matrix equation is being solved in order to obtain {H},+A,, -
-
-
-
[L]{H}t+,,t = {F}t+at,
(24)
where [L] is the modified band matrix on the left-hand side. By using automatic generation of finite elements and numbering the whole generated region with only a small part covering the cross-section S2 of the conductor, some terms of the
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
270
matrices [C] and [D] (corresponding to the points outside the region ~2 of the conductor cross-section) assume zero values. Although the algorithms for solving eqn. (24) are widely known, they are not valid when the diagonal terms of the matrix [L] assume zero values. It is then necessary to put ones in place of the zero terms of the diagonal of [ L] and put zeroes in the columns and rows corresponding to the numbers of those terms (being zero on the diagonal). Moreover, it is also necessary to set to zero the appropriate terms of the vector on the right-hand sides, {F}r+a,. Consequently, we obtain the solutions of eqn. (24) where wc have zero values of the vector {H}t.a, at places lying outside the region ~(2 of the conductor cross-section.
Power losses in the conductor
On dividing the region of the axially-symmetrical conductor into finite elements, the power lost in the whole region O for every time instant is written as a sum of power losses in every element which leads to [1]: 2w
P=--
It ) '(
,, re
OH z + _ _
-~z
r,,
Or
]]
'
where e is the number of the finite element on the conductor region. The other symbols are the same as in eqn (12).
Numerical calculations
Numerical calculations were carried out for an axially-symmetrical bushing with the cross-section presented in Fig. 3. Along its symmetry axis is placed a thin linear conductor carrying current defined by eqn. (1). In order to extend the possibility of applying the numerical calculations once they have been made, the following parameters and relative
Fq l'r'-c/<'-,'!~-;
.rez
"~ w
I I Fig. 3. Cross-section of the bushing.
A. Gqsiorski / Transientelectromagneticfield in a conducting bushing
27l
dimensions were assumed: relative bushing radius; RR = r2/r 1 relative bushing height; HR = h/r 1 relative area of the finite element; T O= T / ( h r 1) dimensionless skin-effect parameter; K R = ~/to/xy r t dimensionless time-step parameter; DT = to At = 2"rrf At dimensionless coordinate r i of the point i; RI = ri/r l ER = (r i + Q + rk)/rl; AP = av/rl, BP = bp/h, p = i, j, k P = I, J, K. After taking these parameters and relative dimensions into account, expressions (9)¢ (10) and (23) take the following forms: 1 [[ A I - A I AI-AJ AI.AK ] [ce]= 4ToH------ffIIA J . A I AJ.AJ AJ.AK [AK.AI AK.AJ AK.AK
J
'I-BI'BIBI + ( H R ) 2_ BJ" [BK'BI
BI-BJ SJ" BJ BK'BJ
Bj.BI'BK]) + - BK 4(ER)2 BK'BK
' 1]
2
1,
1
2
(26) [de]
12
=
2
,
(27)
1 1
Ilk lieS ---= 2"rrRI [sin(k D T + a )
sin a e x p ( - k DT x)],
(28)
where i is the number of the boundary point, k is the number of the next-time-step, and K =/3/to is a factor describing the decay rate of the transient component. After finding the derivatives, intergrating over the finite element region in eqn. (25), comparing the calculated power losses to P0 = 12 2 w ( r 1 + r 2 ) / 2 yhr, Y'. T o '
(29)
e
as well as considering relative dimensions, the instant unit power losses in the bushing take the form: e
Y~'T°
p, = __ =
P0
e
R~ + 1
ER ~ __[~e
(30)
7"0
where:
~e =/_/?(AI 2 + nR2m~) + & ( A j ~ + HR~BJ~) +/_/~(AK2 + HR~BK ~) + 2 [ H i H j ( A I " AJ + HR2BI • BJ) + H j H k ( A J . AK + HR2Bj • BK) + H ~ H I ( A K " AI + HR2BK • BI)] +
dToHR 2 E~ [ H/2BI + H~aBJ + H d B K + H , Hj(BI + BJ) + HjHk(BJ + BK) + H k H i ( B K + BI)]
ToHR 2 + ER----~ [ H i ( H i + Hi) + Hi(Hi + Hk) + H k ( H k + Hi) ] .
(31)
272
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
'i?
t
2&9
l't3
Fig. 4. Finite element discretization of the bushing.
Equation (29) describing P0 can be interpreted as an approximate relation describing the power losses in the ring through which the d.c. current I passes. The rectangular region circumscribing the bushing cross-section (see Fig. 4) was subdivided into 512 triangular linear finite elements, which corresponds to 289 discretization points including 80 points on the cross-section boundary. The width of the symmetrical band of band matrices [C] and [D] is equal to 19. Numerical calculations were carried out for the following parameter values: R R = 1.25: H R = 1.0; D T = 2av 0.005 = 0.03141593; a = ~-/6 = 30 °, K = 1.868; for K -- 150 time-steps. On the basis of the calculations, the time diagrams of power losses were plotted (Fig. 5) for values of the coefficient KR varying from 1 to 6. (Note the different scaling factors on the vertical axis for curves A - C . ) Curves D and E, denoted by broken lines, show approximate diagrams of the current i(oo, ~, t) given by eqn. (1) (shifted on the x-axis by +Ix/2) and the transient component of the current (twice the value shown, sign negative), respectively.
Conclusion
From Fig. 5 it follows that the mean value of the power losses (mean value is considered to be the value about which a steady-state component oscillates) increases as the input factor KR is increased. The latter causes an increase in maximum amplitude (at the time when the transient component exists) of the power loss diagram. Comparison of numerical calculations have been made for a rectangular bushing (filling completely the discretized region in Fig. 4). The mean power losses calculated here were compared with the power loss calculated for small values of KR (weak skin-effect) [14] for the quasi-stationary state. It should be noted that the values obtained here are smaller by 3% than those given in Ref. [14].
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
273
e', it,~,¢,t)
%
4.0
f~,\/ a.~ ~
/
~\
// \
°,t,l/,
,
'
28
5O
YS
/
400
4~5
~
K
Fig. 5. Power losses as a function of time for different values of the skin effect parameter: (A) KR = 1 (scale of the vertical axis: 0.5 x 10-4); (B) KR = 3 (scale: 1 x 10-3; (C) KR = 6 (scale: 1 × 10-2). Curves D and E show the current (eqn. (1)) and its transient component, respectively.
It may, therefore, be stated that the finite element method approach with the single-step method is of great use for analyzing transient electromagnetic field processes in conducting bushings with irregular shapes.
Acknowledgments
The author would like to thank Prof. P. Rolicz of the Polytechnic of Cz~stochowa for the interesting and helpful discussions. The author kindly acknowledges the support of the State Committee for Scientific Research (Warsaw, Poland) via Grant PB-2160/3/91.
References [1] W.R. SMYTHE, Static and Dynamic Electricity, McGraw-Hill, New York, 3rd edn., 1968. [2] R. SmORA, J. Pugczvr~sra and W. LIPlr~sro, "Electromagnetic transient in a conductive bushing and in slots of an induction machine", Arch. Elektrotech. (Warsaw) XXV (4) pp. 887-894, 1976 (in Polish). [3] P.P. SILVESTER and A. KONRAD, "Axisymmetric triangular finite element for the scalar Helmholtz equation", Int. J. Num. Methods Eng. 5, pp. 481-497, 1973. [4] P. RoLicz, "Non-steady state in a conducting ring enclosing a conductor with current", Z. Elektr. Inf. Energietech. 8, pp. 419-426, 1978. [5] P. ROLICZ, "Application of the Bubnov-Galerkin method to analysis of transient field processes", Arch. Elektrotech. (Warsaw) XXV (3), pp. 691-702, 1976 (in Polish).
274
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
[6] P. Roucz, "Use of direct methods of functional analysis for calculation of non-steady electromagnetic phenomenons in electric machines and arrangements", Politech. Poznatlska Rozpr. 85, pp. 1-67, 1977 (in Polish). [7] S. SmORA, J. PURCZVr~SK1 and W. LIpIr~S~, "Die analytische Bestimmung des magnetischen Feldes und der Wirbelstromverluste in der leitenden Kraisscheibe als L6sung des Dirichlete-Problems fiir die Helmholtzsche Gleichung", Arch. Elektrotech. (Berlin)57, pp. 247-251, 1975. [8] P. Roucz, "Eddy currents generated in an elliptic conductors by a transverse alternating magnetic field", Arch. Elektrotech. (Berlin) 68, pp. 423-431, 1985. [9] P.P. SmVESTER and C.R.S. HASt.AM, "Magnetotelluric modelling by the finite element method", Geophys. Prospect. 20, pp. 872-891, 1972. [10] M. HARA, T. WADA, A. TOYOMA and F. KIKUCHI, "Calculation of RF electromagnetic field by finite element method", Sci. Pap. Inst. Phys. Chem. Res. (Jpn.) 75 (4), pp. 143-175, 1981. [11] O.C. ZIENKtEWlC'Z, The Finite Element Method, McGraw-Hill, New York, 3rd edn., 1977. [12] M. ZLAMAL, "Finite element method in heat conduction problems", in: The Mathematics of Finite Elements and Applications, Vol. II, edited by J. WHITEMAN, Academic Press, London, pp. 85-104, 1976. [13] A. K~AWCZYK, "Application of finite element method to transient problems of technical electrodynamics', Rozpr. Elektrotech. 29 (2), pp. 331-344, 1983 (in Polish). [14] J. PU~CZVr~SKI and R. SmonA, "Power losses in a ring enclosing a conductor with alternating current", Arch. Elektrotech. (Warsaw) XXlIl, (3), pp. 695-710, 1974 (in Polish).
265
FINEL 239
Analysis of the transient electromagnetic field in a conducting bushing A l e k s a n d e r K. G~siorski Department of Electrical Engineering, Polytechnic of Czestochowa, Poland
Received January 1987 Revised February 1992 Abstract. The paper presents a general method for the analysis of the transient electromagnetic field and the
calculation of power losses in axially-symmetrical conductors with known boundary conditions. The method consists of the numerical version of the Bubnov-Galerkin method (finite element method) for discretization of the conducting region and the one-step method for discretization of time. On the basis of algebraic considerations, numerical computations for the selected shape of the bushing used in practice have been performed. Calculations of the relative power losses in the bushing as a function of time have been performed and graphs based on them are presented.
Introduction
Let us consider a solid of revolution (axially-symmetrical relative to the revolution axis z ) m a d e of isotropic n o n - f e r r o m a g n e t i c conducting material with constant magnetic permeability /-~ = ~0 and conductivity y (Fig. 1). T h e solid is placed in the field of a thin c o n d u c t o r which, starting at t = 0, carries an alternating current (with its transitory c o m p o n e n t ) described by the relation: i ( t o , / 3 , t) = I v ~ - [ s i n ( t o t + a ) - s i n
a exp(-/3t)],
t~>0,
(1)
where w is the angular frequency, and a , / 3 are constant values. T h e thin c o n d u c t o r is placed in space in such a way that on the surface S of the solid, the magnetic field intensity lip is constant and i n d e p e n d e n t of the angle 0 of the cylindrical system (r, 0, z) [1]. F o r such an axially-symmetrical solid, both the magnetic field in its interior and the inducting field have only one non-zero c o m p o n e n t in the 0-direction. The equation describing this c o m p o n e n t takes the form [2,3]: OH V2H - ].~'y-~"- ~-- 0,
(2)
where t~2H 1 aH 02H V Z H = 0r 2 + -r - ar- + -az 2
H r2'
Correspondence to: Dr. Eng. Aleksander K. G]siorski, Department of Electrical Engineering, Polytechnic of Czestochowa, 17 Armii Krajowej Avenue, Czestochowa, Poland.
0168-874X/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
266
,t-tl 'lv
Fig. 1. Axially symmetrical solid placed in the field of a straight conductor with current.
with the initial condition: H(r,
t : O) I t~ : 0
for the entire region of the wire
(3)
and the Dirichlet boundary condition:
H(r,t)[s=Hp.,{s
for t >~0,
(4)
where S is the boundary of the region g2 of the conductor.
Method of solving the problem The analytical solution of eqn. (2) with conditions (3) and (4) is limited to solids with simple cross-sectional shapes [2,4-6]. For much more complex cross-sectional shapes it is necessary to find the solution numerically [7]. The finite element method is being more and more widely used for discretization of regions with complex cross-sectional shapes. The numerical method used in this paper is based on the direct Bubnov-Galerkin method. This method has been used in electrical engineering since 1973 [4-6,8]. In order to arrive at an approximate solution of eqn. (2) with the conditions (3) and (4), it is assumed that the magnetic field distribution in the conducting solid takes the form [5,6]: 1,
H = S, H~,,
(5)
1-- o
where the Ct are basis functions independent of time, disappearing on the boundary, H t are unknown values, but H0~b0 = i ( w , [3, t)/(2vrr). Therefore function H always satisfies, as required, the Dirichlet boundary condition (see eqns. (23) and (28)) regardless of the values of coefficients H l, 1 ~
A. Gqsiorski / Transientelectromagneticfield in a conductingbushing
267
eqn. (5). By making use of the Green formula [6,8]. 0u fagrad v grad u dO + fay V2u dO = -Is [v-dS'On
(6)
we obtain the Bubnov-Galerkin equations system [5,6] for the diffusion equation (2) with the boundary conditions (3) and (4):
grad q,, grad ~,~ + -~,~m
~ H~ l=1
dO + ~ ' ~
~,~mdO = 0,
(7)
where /~ is the conductor region. In the system of equations (7) integrals on the boundary do not appear because they vanish, since ~om I s = 0 for 1 ~
( [Ce] + --[d 1 e]-'~
Hj = 0, Hk
ca
(8)
where [9,11,12] H i, Hi, H k are the values of the magnetic field intensity at vertices i, j, k,
1 (Iaiai aja ii [Ce]= ~ aka
aiaj ajaj akaj
aiakI Ibibi
aja k + bib i aeak bkbi
11]
12
2 1
1 ; 2
bjbj bkbj
bib k bkbk
+-r2 -~
1
2 1
; (9)
(10)
T is the area of the triangular finite element with coordinates (r i, Zi) , (rj, Zj), (rk, zk) for the vertices, given by T = 7' det
I!
ri rj
Zi]
zi ;
(11)
Fk Zk 1 re = 3(r i + rj + rk);
(12)
ai=rk-r~,
(13)
b i = z i - z k.
The remaining coefficients a j, bi, ak, b,, are determined by cyclic changes of the indices i, j, k. Summing the element equations over all finite elements g2e in the region /2, a global equation is formed for relation (7):
-d[Dl~){tt}
[c] + 1
d
=0.
(14)
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
268
i"! d
r
HJ,__+~,_
H(t)
/
;
J
,
l
i:
t+At
±
Fig. 2. P l o t s o f H(t) a n d d H ( t ) / d t v e r s u s t i m e o v e r t h e t i m e i n t e r v a l t to t + A t .
The matrix equation (14) should be valid for every instant of time, including times t and t + At. For these times, it takes the respective forms: l
d
[ C ] { H } , + ~ [ D l d - 7 ( H }, = 0, 1
15)
d
[ C ] { H } , + ~ , + - - [ O ] - d - ; { H } , + . , , = 0. ¢0
16)
Let us assume that in a small time interval A t, the nonlinear continuous function f = d H(t)/d t can be linearly interpolated and approximated by a straight line linking the point Pl(t)= f(t) and P2(t + At)=f(t + At). These points are placed at the begining and end of the time interval (Fig. 2). The straight line can be represented by d
~ H ( t) =at + b,
(17)
where a and b are constant coefficients. From eqn. (17)
f(m'+~'dH= f'~'(at + b) dr.
(18)
{tt}t
After integration, we arrive at {HI t+At
-{U}t=½a[(At+t
) 2 __
t z] + b ( t + A t - t ) .
(19)
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
269
From eqn. (17) d d -d--t-{H}t+at + -~f{H}t=a[(At+t+t) + 2 b ] .
(20)
On comparing the last two expressions, we can write d d 2 ~-7 {H},+a, + ~-7{H}, = ~-7({HIt+a, - {H},).
(21)
After equating the expressions (15) and (16) and using the relation (21), and performing some algebraic transformations, we obtain ( l~__A_7[D] +
½[C]){H}t+at =
( l~_A_7[D] _
½[C]){H}t.
(22)
The procedure adopted above corresponds to a more general analysis provided in [13] and it is called the Crank-Nicholson scheme. In order to find the value of the v e c t o r {H}t+At at the next time step, it is necessary to know its preceding value {H},. For the start value (t = 0) of the vector {H}0 we can simply take the zero value. In the course of the calculation process, the matrices [C] and [D] dependent on the geometry and parameters of the conductor material do not change, which makes it possible to calculate them only once. This reduces the time of calculation. The numerical values of the boundary condition (4), which change in the course of the numerical process and are different for every time instant considered, can be represented on the basis of the flux law [1] and eqn. (1) in the form: 1
lip. t Is = -~-R[sin(oot+a)
-sin a exp(-flt)],
peS,
(23)
where R is the radial distance of boundary point p (see Fig. 1). The other symbols are the same as for eqn. (1). The boundary conditions at every time step will be incorporated into the currently solved system of equations (22). The matrix equation (22) appears to be easy to solve numerically. The numerical procedure, however, is complicated by the necessity of considering the time-varying values of the boundary condition. The procedure for every time step is as follows: The vector {H}t+a, is filled with the boundary values corresponding to the time step t + At. The left-hand side of expression (22) is calculated with the known values of {H}t determined in the previous step, which--after multiplying--is defined as {Fi}t+at. - The right-hand side of expression (22) is calculated and set equal to the vector {Fr}' with {H}t determined in the previous step. The terms of the vector {F}t+at on the right-hand side are determined by subtracting the vector {Fi}t+at from {Fr}t, and then substituting the terms corresponding to the known values of boundary conditions, given in terms of the vector {H}t+a,. - For the vector {H}t+at only those terms placed on the boundary have known values. For every boundary term, it is necessary to modify the left-hand side matrix (1/to At)[D] + ½[C]. To do that, we put zeroes in both the row and column relating to the considered boundary term of the vector {H}t+at and put a value of one in the place of their intersection. Finally the matrix equation is being solved in order to obtain {H},+A,, -
-
-
-
[L]{H}t+,,t = {F}t+at,
(24)
where [L] is the modified band matrix on the left-hand side. By using automatic generation of finite elements and numbering the whole generated region with only a small part covering the cross-section S2 of the conductor, some terms of the
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
270
matrices [C] and [D] (corresponding to the points outside the region ~2 of the conductor cross-section) assume zero values. Although the algorithms for solving eqn. (24) are widely known, they are not valid when the diagonal terms of the matrix [L] assume zero values. It is then necessary to put ones in place of the zero terms of the diagonal of [ L] and put zeroes in the columns and rows corresponding to the numbers of those terms (being zero on the diagonal). Moreover, it is also necessary to set to zero the appropriate terms of the vector on the right-hand sides, {F}r+a,. Consequently, we obtain the solutions of eqn. (24) where wc have zero values of the vector {H}t.a, at places lying outside the region ~(2 of the conductor cross-section.
Power losses in the conductor
On dividing the region of the axially-symmetrical conductor into finite elements, the power lost in the whole region O for every time instant is written as a sum of power losses in every element which leads to [1]: 2w
P=--
It ) '(
,, re
OH z + _ _
-~z
r,,
Or
]]
'
where e is the number of the finite element on the conductor region. The other symbols are the same as in eqn (12).
Numerical calculations
Numerical calculations were carried out for an axially-symmetrical bushing with the cross-section presented in Fig. 3. Along its symmetry axis is placed a thin linear conductor carrying current defined by eqn. (1). In order to extend the possibility of applying the numerical calculations once they have been made, the following parameters and relative
Fq l'r'-c/<'-,'!~-;
.rez
"~ w
I I Fig. 3. Cross-section of the bushing.
A. Gqsiorski / Transientelectromagneticfield in a conducting bushing
27l
dimensions were assumed: relative bushing radius; RR = r2/r 1 relative bushing height; HR = h/r 1 relative area of the finite element; T O= T / ( h r 1) dimensionless skin-effect parameter; K R = ~/to/xy r t dimensionless time-step parameter; DT = to At = 2"rrf At dimensionless coordinate r i of the point i; RI = ri/r l ER = (r i + Q + rk)/rl; AP = av/rl, BP = bp/h, p = i, j, k P = I, J, K. After taking these parameters and relative dimensions into account, expressions (9)¢ (10) and (23) take the following forms: 1 [[ A I - A I AI-AJ AI.AK ] [ce]= 4ToH------ffIIA J . A I AJ.AJ AJ.AK [AK.AI AK.AJ AK.AK
J
'I-BI'BIBI + ( H R ) 2_ BJ" [BK'BI
BI-BJ SJ" BJ BK'BJ
Bj.BI'BK]) + - BK 4(ER)2 BK'BK
' 1]
2
1,
1
2
(26) [de]
12
=
2
,
(27)
1 1
Ilk lieS ---= 2"rrRI [sin(k D T + a )
sin a e x p ( - k DT x)],
(28)
where i is the number of the boundary point, k is the number of the next-time-step, and K =/3/to is a factor describing the decay rate of the transient component. After finding the derivatives, intergrating over the finite element region in eqn. (25), comparing the calculated power losses to P0 = 12 2 w ( r 1 + r 2 ) / 2 yhr, Y'. T o '
(29)
e
as well as considering relative dimensions, the instant unit power losses in the bushing take the form: e
Y~'T°
p, = __ =
P0
e
R~ + 1
ER ~ __[~e
(30)
7"0
where:
~e =/_/?(AI 2 + nR2m~) + & ( A j ~ + HR~BJ~) +/_/~(AK2 + HR~BK ~) + 2 [ H i H j ( A I " AJ + HR2BI • BJ) + H j H k ( A J . AK + HR2Bj • BK) + H ~ H I ( A K " AI + HR2BK • BI)] +
dToHR 2 E~ [ H/2BI + H~aBJ + H d B K + H , Hj(BI + BJ) + HjHk(BJ + BK) + H k H i ( B K + BI)]
ToHR 2 + ER----~ [ H i ( H i + Hi) + Hi(Hi + Hk) + H k ( H k + Hi) ] .
(31)
272
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
'i?
t
2&9
l't3
Fig. 4. Finite element discretization of the bushing.
Equation (29) describing P0 can be interpreted as an approximate relation describing the power losses in the ring through which the d.c. current I passes. The rectangular region circumscribing the bushing cross-section (see Fig. 4) was subdivided into 512 triangular linear finite elements, which corresponds to 289 discretization points including 80 points on the cross-section boundary. The width of the symmetrical band of band matrices [C] and [D] is equal to 19. Numerical calculations were carried out for the following parameter values: R R = 1.25: H R = 1.0; D T = 2av 0.005 = 0.03141593; a = ~-/6 = 30 °, K = 1.868; for K -- 150 time-steps. On the basis of the calculations, the time diagrams of power losses were plotted (Fig. 5) for values of the coefficient KR varying from 1 to 6. (Note the different scaling factors on the vertical axis for curves A - C . ) Curves D and E, denoted by broken lines, show approximate diagrams of the current i(oo, ~, t) given by eqn. (1) (shifted on the x-axis by +Ix/2) and the transient component of the current (twice the value shown, sign negative), respectively.
Conclusion
From Fig. 5 it follows that the mean value of the power losses (mean value is considered to be the value about which a steady-state component oscillates) increases as the input factor KR is increased. The latter causes an increase in maximum amplitude (at the time when the transient component exists) of the power loss diagram. Comparison of numerical calculations have been made for a rectangular bushing (filling completely the discretized region in Fig. 4). The mean power losses calculated here were compared with the power loss calculated for small values of KR (weak skin-effect) [14] for the quasi-stationary state. It should be noted that the values obtained here are smaller by 3% than those given in Ref. [14].
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
273
e', it,~,¢,t)
%
4.0
f~,\/ a.~ ~
/
~\
// \
°,t,l/,
,
'
28
5O
YS
/
400
4~5
~
K
Fig. 5. Power losses as a function of time for different values of the skin effect parameter: (A) KR = 1 (scale of the vertical axis: 0.5 x 10-4); (B) KR = 3 (scale: 1 x 10-3; (C) KR = 6 (scale: 1 × 10-2). Curves D and E show the current (eqn. (1)) and its transient component, respectively.
It may, therefore, be stated that the finite element method approach with the single-step method is of great use for analyzing transient electromagnetic field processes in conducting bushings with irregular shapes.
Acknowledgments
The author would like to thank Prof. P. Rolicz of the Polytechnic of Cz~stochowa for the interesting and helpful discussions. The author kindly acknowledges the support of the State Committee for Scientific Research (Warsaw, Poland) via Grant PB-2160/3/91.
References [1] W.R. SMYTHE, Static and Dynamic Electricity, McGraw-Hill, New York, 3rd edn., 1968. [2] R. SmORA, J. Pugczvr~sra and W. LIPlr~sro, "Electromagnetic transient in a conductive bushing and in slots of an induction machine", Arch. Elektrotech. (Warsaw) XXV (4) pp. 887-894, 1976 (in Polish). [3] P.P. SILVESTER and A. KONRAD, "Axisymmetric triangular finite element for the scalar Helmholtz equation", Int. J. Num. Methods Eng. 5, pp. 481-497, 1973. [4] P. RoLicz, "Non-steady state in a conducting ring enclosing a conductor with current", Z. Elektr. Inf. Energietech. 8, pp. 419-426, 1978. [5] P. ROLICZ, "Application of the Bubnov-Galerkin method to analysis of transient field processes", Arch. Elektrotech. (Warsaw) XXV (3), pp. 691-702, 1976 (in Polish).
274
A. Gqsiorski / Transient electromagnetic field in a conducting bushing
[6] P. Roucz, "Use of direct methods of functional analysis for calculation of non-steady electromagnetic phenomenons in electric machines and arrangements", Politech. Poznatlska Rozpr. 85, pp. 1-67, 1977 (in Polish). [7] S. SmORA, J. PURCZVr~SK1 and W. LIpIr~S~, "Die analytische Bestimmung des magnetischen Feldes und der Wirbelstromverluste in der leitenden Kraisscheibe als L6sung des Dirichlete-Problems fiir die Helmholtzsche Gleichung", Arch. Elektrotech. (Berlin)57, pp. 247-251, 1975. [8] P. Roucz, "Eddy currents generated in an elliptic conductors by a transverse alternating magnetic field", Arch. Elektrotech. (Berlin) 68, pp. 423-431, 1985. [9] P.P. SmVESTER and C.R.S. HASt.AM, "Magnetotelluric modelling by the finite element method", Geophys. Prospect. 20, pp. 872-891, 1972. [10] M. HARA, T. WADA, A. TOYOMA and F. KIKUCHI, "Calculation of RF electromagnetic field by finite element method", Sci. Pap. Inst. Phys. Chem. Res. (Jpn.) 75 (4), pp. 143-175, 1981. [11] O.C. ZIENKtEWlC'Z, The Finite Element Method, McGraw-Hill, New York, 3rd edn., 1977. [12] M. ZLAMAL, "Finite element method in heat conduction problems", in: The Mathematics of Finite Elements and Applications, Vol. II, edited by J. WHITEMAN, Academic Press, London, pp. 85-104, 1976. [13] A. K~AWCZYK, "Application of finite element method to transient problems of technical electrodynamics', Rozpr. Elektrotech. 29 (2), pp. 331-344, 1983 (in Polish). [14] J. PU~CZVr~SKI and R. SmonA, "Power losses in a ring enclosing a conductor with alternating current", Arch. Elektrotech. (Warsaw) XXlIl, (3), pp. 695-710, 1974 (in Polish).