Energy conversion by nonlinear permanent magnet

machinezi

R.J. Strahan

Indexing terms. Energy conwrsion, Permanent magnet machines, Coupling theory, Reluctance torque, Electromagnetics

residual magnetism. By addressing the magnetisation

Abstract: Stored energy and coenergy are defined process it shows how stored energy may be defined in a
for a permanent magnet system. It is shown that permanent magnet system. By then examining energy
either stored energy or coenergy may be used to methods, a solid theoretical base for a selection of
determine permanent magnet reluctance torque torque equations used by both machine and CAD sys-
where the magnetisation characteristics of regions tem designers, as well as some less obvious equations,
within the system ,are arbitrary. It is shown how is provided. (In this paper, equations for force may be
residual magnetism may be incorporated into obtained from torque equations by replacing the rota-
classical electromechanical coupling theory. It is, tional displacements with linear displacements.)
therefore, shown how general equations for
torque can be derived for nonlinear permanent 2 Energy stored in a permanent magnet system
magnet systems from classical electromechanical
coupling theory. ‘The approximation made in In classical electromechaiiical coupling theory stored
deriving a simplified equation for torque in a energy is a physical quantity which can be measured
linear system is deslxibed. Finally, the validity of experimentally. The stored energy is the energy which
the first quadrant representation of the rate of can be transferred to or from a conservative electrome-
change of coenergy within a permanent magnet chaiiical coupling field via mechanical or electrical ter-
material, relevant to CAD systems on minals. In this Section the definition of stored energy
electromagnetics, is demonstrated. extended to a system exhibiting significant residual
magnetism or permanent magnetism remains essentially
the same. The specification of a conservative electrome-
chanical coupling field thus excludes hysteresis from
1 Introduction the calculation of torque.

Energy methods are widely used and well understood

for determining the torque or force in magnetically
nonlinear machines that do not contain permanent
electrical
terminal
+
-
1 7 1I
”
i

h=h( i )
I svstem consistina- of
- a winding
-hard magnetic material
-soft magnetic material
-airgap &ion and/or linear material I
magnets. Energy methods are employed to calculate
torques or forces of magnetic origin after determina- Fig. 1 Electrical terminal pair representation of U permanent mugnet
tion of the energy stored in the electromechanical cou- system
pling field. The origins of this theory date back at least
as far as [l] where the equation for the force resulting Fig. 1 shows a representation of a permanent magnet
from the ‘mechanical action between two circuits’ in system consisting of a winding and a hard magnetic
the absence of magnetic material is expressed in terms material. The winding has a flux linkage h and current
of currents and inductance coefficients. The scope of i and its terminals are depicted in Fig. 1. An airgap or
this analysis was extended in [2] to provide general linear region and a soft magnetic material may also be
equations for an arbitrary number of circuits which included in the system, The soft magnetic material is
may contain iron, either saturated or not, but are modelled as being anhysteretic with the B-H character-
assumed to have no hysteresis. This has been followed istic passing through the origin. Energy may be trans-
by comprehensive treatments of electromechanical cou- ferred to the system electrically or mechanically. TO
pling theory [3-51. The increasing use and improving simplify the calculation of energy transferred to the sys-
technology of permanent magnet materials has gener- tem, the energy transferred to the system is accounted
ated a need to incorporate materials exhibiting residual for electrically. This is achieved by treating the hard
magnetism into this theory. The purpose of this paper magnetic material as being initially unmagnetised such
is to show how the classical theory can accommodate that initially h = 0 when i = 0 and any forces or tor-
0IEE, 1998
ques of magnetic origin are zero. All frictional and
IEE Proceedings online no. 19981863
resistive losses excluding hysteresis loss are modelled
externally to the system. The system may therefore be
Paper first received 23rd Septemb,:r 1997 and in revised form 5th January
1998 nonconservative during the magnetisation process. The
The author is with the Department of Electrical and Electronic Engineer- system is first mechanically assembled with h held at
ing, University of Canterbury, Private Bag 4800, Christchurch, New Zea- zero and the mechanical energy transferred to the sys-
land tem is zero. The flux linkage is then raised from zero
IEE Proc-Electr. Power Appl., Vol. .’45,No. 3, Muy 1998 193
and a voltage e = dWdt is induced across the electrical is nonrecoverable because the magnetisation character-
terminals by the magnetic field. The energy transferred istic cannot be retraced back to B = 0 at H = 0 from
is obtained, in this case, by the classical equation for within the first quadrant. The recoverable energy will
stored energy in a singly excited system: be defined as the ‘stored energy’. At € =I 0 with B = B,
no more energy is recoverable and the stored energy is
19 = .I, idX
rX
zero. (Note: After completion of a full cycle of a hys-
teresis loop, the magnetisation is returned its original
The energy transferred is absorbed as energy which is condition, and nonrecoverable energy has been dissi-
recoverable and also as energy which is not recovera- pated as heat called the hysteresis loss [7].Similarly, if
ble. However, eqn. 1 and the A - i characteristic do not, the B-H characteristic in Fig. 2a is extended into the
in general, provide sufficient information to allow the 2nd and 3rd quadrants such that a hysteresis loop is
components of recoverable and nonrecoverable energy compeleted returning to B = H = 0, the nonrecoverable
to be determined. Eqn. 1 is equivalently expressed in energy of the first quadrant has been dissipated as hys-
terms of the energy density of the magnetic field corre- teresis loss.)
sponding to vectors B and H integrated over the vol- The flux density is now reduced to B, by a demag-
ume of the system by netising field H, during which H.dB is positive and
energy corresponding to the areas of both shaded
regions in the second quadrant is absorbed. The
8=hlBH.d13dv (2) demagnetising field is now reduced to zero and it is
assumed that a minor hysteresis loop is followed to Bo.
This mathematical transformation is described in [6], The hysteresis loss in cycling between N = 0 and H, is
pp. 122-124. The field may be due to both currents and assumed to be small such that the minor loop can be
residually magnetised material. Eqn. 2 allows the approximated by a recoil line. Therefore upon initially
energy transferred to the system to be separated using reaching Bo, the darker shaded area in the second
B-H characteristics into components within elements of quadrant corresponds to nonrecoverable energy, and
the system volume as follows. Fig. 2a shows a B-H the lighter shaded area to stored energy returned to the
characteristic for a hard magnetic material. From B = electrical terminals or absorbed by some other region
0 the characteristic follows the initial magnetisation or both. For subsequent movement of the operating
curve until the saturation flux density B,, is reached. point along the recoil line, or as long as the characteris-
B t tic remains single-valued within the limits of integration
Bo to B,,, hysteresis is excluded and the permanent

JI.. cm
magnet stored energy is given by

wm= H, * dB,du, (3)

Fig. 26 and c show B-H characteristics for a single-val-
ued soft magnetic material, and air or linear material,
H respectively. The areas of the shaded regions corre-
spond to stored energies. Given that the hard magnetic
material has reached a single-valued state within the
limits described above, the electromechanical coupling
field is conservative, and the stored energy of the per-
E t B t manent magnet system is given by

H, dB,dv,

+ kslBs H, . dB,du,

I, (4)
0 H, H 0 Ha H
b C The inner integrals of the three RHS terms of eqn. 4
are the energy density functions of the permanent mag-
Fig.2 B H characteristics and energy densities net, soft material, and linear material, respectively.
a Hard magnetic material
b Soft magnetic material Some examples of these energy density functions are
c Air or linear matcrial
given in [8, 91.
An equation is given in [lo] where the stored energy
The energy density corresponding to energy absorbed of the permanent magnet system is calculated by inte-
by this magnet region is depicted by both shaded areas grating over only the volume of the magnet using
in the first quadrant. The field intensity H is then
reduced to zero and the flux density follows the major [H, dB, - B, . dH,]d~,
hysteresis curve from B,, to B, in which H.dB is nega- w=:/J *
(5)
tive and recoverable energy is returned to the electrical Eqn. 5 is exactly equivalent to
terminals or absorbed by some other region or both.
The recoverable energy corresponds to the lighter H,. dB,dv, -
shaded area in the first quadrant. The darker shaded
area corresponds to nonrecoverable energy. This energy
194 IEE Pvoc -Electv Power Appl, Vol. 145, No. 3, May 1998
From eqn. 39 in the Appendix (Section 9.1) it can be
shown that
1
B, . lI,d~, z= B, H , ~ v , ( 7 )
+I f lHs B, . dH,dv,

if all currents are zero. Eqns. 5 and 6 are, therefore,

+ -'j
1
.i,
Ba. Had,% (12)
only valid if the region outside the permanent magnet For a permanent magnet system, with the winding de-
is linear and all currents are zero. energised, eqn. 41 in the Appendix (Section 9.1) shows
Expressions for B and H may be derived as functions that
of electrical and mechanical terminal quantities. Eqns. W ' ( i = 0) = -W(i = 0)
4-6 may therefore be expressed as functions of rota- (13)
tional displacements for the calculation of reluctance Substitution of eqn. 13 into eqn. 8 shows that the reluc-
torque. For a rotational displacement 6' with the wind- tance torque is obtained in terms of coenergy by
ing de-energised, or removed, the resulting reluctance
torque is defined as the negative rate of conversion of
stored energy into mechanical energy:
d W ( i = 0) 4 Electromechanical coupling
T,€l = -
d0
The definition of stored energy given here yields 4.1 Permanent magnets and single
expressions for stored energy which, when used in en e rgised win ding
eqn. 8, are shown to give accurate values of permanent Fig. 1 is now extended to include a mechanical terminal
magnet reluctance torque [8, 91. such that simultaneous electrical and mechanical
The definition of stored energy provided here permits energy conversion may occur. If current and flux link-
determination of the relationship of the mathematical age are now defined to be state functions then hystere-
quantity 'coenergy' to stored energy where all currents sis is excluded, the functional relationship between
are zero, in Section 3. Energy methods are examined these variables is single-valued, and the system is con-
more generally in Sections 4.1 and 4.2 to include servative [3]. The conservation of power may then be
nonzero currents. described by
dW - dX
- - 2-
d0
- T-
3 Coenergy of a permanent magnet system (15)
dt dt dt
For the system described in Section 2, the transferred such that the differential energy is given by
coenergy may be deter,mined from dW(X,e) = idX - Td6' (16)
it' =
Eqn. 9 is equivalently expressed by
iz Ad2 (9)
whereby the torque is obtained in terms of stored
energy by the classical result:

H
Ilf=// B.dHdv (10) where the partial derivative is taken with h held con-
v o stant. The differential energy dW(h, @)must have the
After completing the magnetising sequence described
in Section 2, the coenergy density corresponding to H properties of a state function for eqn. 16, and thus
= H, is shown by the shaded area in Fig. 3a. As long eqn. 17, to hold. However, this does not imply that
dW(h, 0) or W(h,0) are required to have the properties
Bc B c of state functions for all values of independent varia-
bles h and 6'. This imposes the constraint that if any of
the independent variables are outside of a range where
W(h, 0) has the properties of a state function, then the
torque cannot be obtained using eqn. 17 for those val-
ues of the independent variables.
a b C The stored energy is obtained by integration of
Fig.3 Coeizergy densities eqn. 16:
U Hard magnetic material
b Soft magnetic material
c Air or linear material W ( X , 0) = 1;' idX - T d 0 (18)

as the demagnetising field remains within limits in The line integral of eqn. 18 is simplified by assembling
which the characteristic remains single-valued, the per- the system by the method described in Section 2 such
manent magnet coenergy is given by that the energy transferred to the system in raising the
flux linkage to a final value is given by eqns. 1 and 2.

wA= LHnk
Lvrz B, . d H , d ~ ,
Fig. 3b and c show the areas corresponding to coenergy
(11) To determine the stored component of transferred
energy, eqn. 2 must be used. In raising the flux density
of the magnet from zero to Bo, the system is not con-
for a single-valued soft magnetic material and air or servative. However, because the magnetisation history
linear material, respectivdy. The coenergy of the per- is known, the stored energy can be calculated within
manent magnet system is given by these limits, and is found to be zero. Therefore, the

W' = Lr"lH''? B . dH,, dum

stored energy is obtained by raising the flux density of
the magnet from Bo to B,, through which the stored
energy is regarded to have the properties of a state
IEE Pvoc -Electr Power A p p l , Vol 145, No 3 May 1998 195
function and is given by eqn. 4. The stored energy is shows that the relationship
regarded to have the properties of a state function if J
the demagnetising field H, remains within limits such
that the demagnetising characteristic remains single-
w + W‘ = A,%, (27)
3=1
valued. The state function requirement of eqn. 16 is
holds for a permanent magnet system with multiple
therefore satisfied allowing the torque of the permanent
magnet system to be given by eqn. 17. energised windings. Application of eqn. 27 allows the
In a conservative system the relationship between torque to be equivalently expressed by
energy and coenergy is given by a Legendre transfor-
mation:
W‘=Xz-W (19)
which is necessarily shown in the Appendix (Section
9.2) (by setting J = 1) to hold for a permanent magnet
system. This relationship allows the torque to be equiv-
alently expressed by the remaining classical results:

where (i, 0) is an abbreviation for (il, ..., iJ; 01, ..., OK).
aw’(2,0) With CAD packages on electromagnetics, it is known
T= that values of torque can be accurately determined
a0 from the rate of change of the total coenergy computed
by integrating the coenergy density over the volume of
the system, as demonstrated by [ll]. This confirms the
validity of eqn. 29. The validity of torque eqns. 24, 28
where the partial derivatives of eqns. 21 and 22 are and 30 are demonstrated by mathematical equivalence
taken with i held constant. The coenergy W for a per- to eqn. 29, resulting from the proof of eqn. 27 given
manent magnet system is obtained by eqn. 12. Eqns. in the Appendix (Section 9.2). (Note: The ‘work func-
17, 20-22 each allow the torque to be obtained for a tion’ formulation in [9] is equivalent to eqn. 30, and is
nonlinear permanent magnet system. For these equa-
supported numerically by comparison to Maxwell
tions, it is essential to hold the independent variable h
stress results.)
or i constant while taking the partial derivative analyti-
In many CAD packages, a representation of perma-
cally or numerically. Note that eqns. 17 and 21 are nent magnets described in Section 6 is useful for
more general forms of eqns. 8 and 14.
numerical computation. This representation is shown in
Section 6 to give an identical rate of change of perma-
4.2 Permanent magnets and multiple
nent magnet coenergy to that of the second quadrant
energised windings
Fig. 1 is now extended to include J electrical and K representation of coenergy given in Section 3 by
mechanical terminal pairs. The energy differential is eqn. 11. Thus coenergy, as defined by eqn. 12, is also
then given by shown by equivalence to yield accurate values of
torque.

5 Torque equations for a linear permanent

j=1 k=l magnet system
whereby the torque obtained at the kth mechanical ter-
minal is obtained in terms of stored energy by In the absence of iron saturation, where a single wind-
ing is energised, the flux linkage of the winding may be
given by
x = Amp) L ( Q + (31)
where (A,0) is now an abbreviation for (Al, ..., hJ; 01, where Am is the flux linkage due to the magnet, and L
..., OK). If the system is assembled in an analogous is the inductance of the winding. Substituting eqn. 31
manner to that described in Section 2, the energy trans- into eqn. 22 yields
ferred to the system in raising the flux linkages to their
final values is given by eqn. 2 and also by

The stored energy W(i, e) is determined by eqn. 4.

”U,...,U 3=1 Fig. 2a shows that if B, increases towards Bo due to,
where iJ = $(A,,..., AJ; O,, ..., &I. If there is no hard for example, a winding current increase, the stored
magnetic material in the system and the functional rela- energy of the magnet decreases. A corresponding
tionships between variables is single-valued, eqn. 25 increase in B in a region surrounding the magnet yields
obtains the stored energy as a state ftinction given that an increase in the stored energy in that region. Eqn. 32
independence of path is demonstrated by satisfying the can be simplified by approximating the energy stored
following equalities: to correspond to mutually exclusive components pro-
vided by the winding and the magnet, whereby
+
TW = ~ (= 0i , 0) 1/2 L ( B ) ~ ~ (33)
(26) and the torque is approximated by
If a permanent magnet is present, the stored energy is 1 .,dL dW(i = O , d )
determined by eqn. 4. The Appendix (Section 9.2)
T=~-+-Z -- (34)
d0 2 dQ do
196 IEE Proc.-Electr. Power Appl., Vol. 145, No. 3, May 1998
which is given in [12] and is shown to model the a constant. The rate of change of coenergy is therefore
motion of a single phase permanent magnet motor suf- the same for both representations, thus yielding identi-
ficiently accurately in [13]. The first term in eqn. 34 is cal values of torque. This relationship provides first: a
used to calculate the torque due to the coupling supporting theoretical basis for the first quadrant rep-
between a magnet and an energised winding in brush- resentation; and secondly supporting evidence for the
less permanent magnet machines. The remaining two experimental validity of the second quadrant represen-
terms describe the 1orques obtained due to reluctance tation.
variation with rotational displacement. Eqn. 34 is par- However, caution must be observed if stored energy
ticularly useful for experimental purposes because all of rather than coenergy is used, as the respective first and
the quantities can be measured from electrical and second quadrant rates of change o€ stored energy are
mechanical terminals. different. In this case, only a second quadrant represen-
6 Current sheet model of a permanent magnet tation has a theoretical basis.

In a permanent magnet material the relation of B to H 7 Conclusions

may be expressed in the form of [6], pp. 13, 129:
Stored energy and coenergy have been defined for a
B = PO[H + M(H, MO) +MO] (35) permanent magnet system. It has been shown that
M is the induced polarisation de€ined by M = xmH either stored energy or coenergy may be used to deter-
where magnetic susceptibility xm is defined by xnl = mine permanent magnet reluctance torque where the
dM/dH. MO is the residual magnetisation which is magnetisation characteristics of regions within the sys-
nonzero in permanent magnet regions such that B is tem are arbitrary. It has also been shown how residual
nonzero when H = 0. M, is interpreted as a source of magnetism may be incorporated into classical electro-
the field. MOmay be replaced by a stationary volume mechanical coupling theory. It has therefore been
distribution of current throughout the volume of the shown how general equations for torque can be derived
magnet of density for nonlinear permanent magnet systems from classical
J = curlMO (36) electromechanical coupling theory. In doing this it has
and with a current distribution on the surface bound- been shown that the relationship W + W‘ = Ai holds
ing the magnet volume of density for a permanent magnet system. The approximation
K = MOx 11 made in deriving a simplified equation for torque in a
(37) linear system has been described. Finally, the validity
where n is the unit outward normal to the surface ([6], of the first quadrant representation of the rate of
p. 129). With MO replaced by an equivalent current change of coenergy within a permanent magnet mate-
sheet, eqn. 35 reduces to B = po[H + MI which
describes a B-H characteristic of the first quadrant rial, relevant to CAD systems, has been demonstrated.
passing through the origin. The shifted curve represen-
tation is shown in Fig. 4b. 8 References
I t3 BI
1 MAXWELL, J.C.: ‘A treatise on electricity and magnetism’, vol.
2 (republished Dover, New York, 1954, 3rd edn.), Art. 583
2 DOHERTY, R.E., and PARK, R.H.: ‘Mechanical force between
electric circuits‘, Trans. AZEE, 1926, 45; pp. 240-252
3 WHITE, D.C., and WOODSON, H.H.: ‘Electromechanical
energy conversion’ (J. Wiley & Sons, New York, 1959)
4 FITZGERALD, A.E., KINGSLEY, C., and UMANS, S.D.:
‘Electric machinery’ (McGrdw-Hill, New York, 1992, 5th edn.)
5 WOODSON, H.H., and MELCHER, J.R.: ‘Electromechanical
dynamics. Part I: Discrete systems’ (J. Wiley & Sons, New York,
a b 1968)
Fig.4 Representations of ptrmrznent magnet coenergy density 6 STRATTON, J.A.: ‘Electromagnetic theory’ (McGraw-Hill, New
a Second quadrant demagnetisation curve York, 1941)
b Curve shifted to first quadrant
7 CHIKAZUMI, S.: ‘Physics of magnetism’ (J. Wiley & Sons, New
York, 1964), p. 17
CAD packages on electromagnetics use the technique 8 HOWE, D., and ZHU, Z.Q.: ‘Influence of finite element discreti-
sation on the prediction of cogging torque in permanent magnet
of shifting the second quadrant demagnetisation curve excited motors’, ZEEE Truns. Magn., 1992, 28, (2), pp. 1080-1083
to the origin and introduce a suitable current carrying 9 MARINESCU, M., and MARINESCU, N.: ‘Numerical compu-
coil for modelling a permanent magnet [14]. The torque tation of torque in permanent magnet motors by Maxwell stresses
may be obtained from the current sheet mode1 using and energy method’, IEEE Trans. Magn., 1988, 24, (1), pp. 463-
466
Maxwell stress [I41 or some other method. 10 ZIJLSTRA, H.: ‘Permanent magents; theory’ in WOHLFARTH,
With CAD package:;, it is known that values of E.P. (Ed.): ‘Ferromagnetic materials’, vol. 3 (North-Holland,
torque can be accurately determined from the rate of 1982), chapter 2
11 BRAUER, J.R., LARKIN, L.A., and OVERBYE, V.D.: ‘Finite
change of the total coenergy using the first quadrant element modelling of permanent magnet devices’, J. Appl. Phys ,
representation of permanent magnet coenergy, as dem- 1984, 55, (6), pp. 2183-2185
onstrated by [Ill. Coenergy density for the first quad- 12 KAMERBEEK, E.M.H.: ‘Electric motors’, Philips Techn. Rev.,
1973, 33, (819). pp. 215-234
rant representation of a permanent magnet is shown by 13 SCHEMMANN,. H.: ‘Theoretische und experimentelle untersuc-
the shaded region in Fig. 4b. The relationship between hungen uber das Dynamische verhalten eines Einphasen-syn-
first and second quadrant coenergy representations is chron-motors mit dauermagnetischem laufer’. PhD thesis,
Tcchnischc Hogeschool, Eindhoven, Oct. 1971
given by 14 GUPTA, R., YOSHINO, T., and SAITO, Y.: ‘Finite element
U>; == w; WiH, + (38) solution of permanent magnetic field’, IEEE Trans. Mugn., 1990,
26, (2), pp. 383-386
where w)2 is a negative coenergy density. The term w \ ~ ~ 15 BROWN, Jr. W.F.: ‘Magnetostatic principles in ferromagnetism’
is the area under the shifted curve from 0 to H, and is (North-Holland, 1962), pp. 44-45

IEE Proc -Electr Power Appl, Vol 145, No 3 May 1998 191
9 Appendix Fig. 5 . Unit normal vector nB, B, and dl are parallel. 4
is the integrated flux which will be required to remain
9. ‘I constant wherever eqn. 44 is evaluated over contour C.
A quasistatic permanent magnet system with all wind- The path chosen for C ensures that B . n # 0 such that
ings de-energised satisfies curl H = 0 and div B = 0, I$ # 0. Multiplying eqn. 42 by eqn. 44 gives
from which it can be shown that [lo, 151

.If H . Bdv = 0 (39)

B . n ~ d a H~ . dl = q3 kJ Jf . n ~ d a ~(45)
The LHS of eqn. 45 is transformed as follows. Let $cH
where V is volume of the permanent magnet system. By . dl = X E l Hi . dl, = XLZIHi nBiAZi, giving
applying the rule of differentiation, whereby d(H . B) = 00
H . dB + B &, eqn. 39 is expressed as

[/H.dB+/B.dH] dv=O (40)

B ,n B d a B
h .ng,U,)(Bi.nB;Aag; )
H . d l = x(Hi
i=l
(46)
which according to eqiis. 4 and 12 may be written as which is equal to
W(2= 0)+ W’(i = 0 ) = 0
W

(41)
where the magnetisation characteristics of regions
E(
i=l
\Hi\InB71 Cos Y H ,) ( IBt I I COS YB;)a’& (47)
within the system are arbitrary. where Avi = AliAaB,is an element of volume. Eqn. 47 is
, contour C simplified by letting ln,j = 1 and cos yBi = 1 to yield

5
z= 1
cos~~,Avi
JHiJIBiJ (48)

yHi is also the angle between Hi and B , therefore

00 00

surface S,

= Lc
z=1

H . Bdvc (49)

V , is the filament volume corresponding to contour C.

9.2 The RHS of eqn. 45 is transformed as follows. The
The permanent magnet system of Section 9.1 is now
extended to the case where curl H = Jfand div B = 0, integral JsSr, nJdaJ may be expressed as a sum of con-
where Jj is the free current density which will be attrib- tributions from winding currents crossing surface SJ by
uted to winding currents. The integral forms of the ZJ”=ly/i, where i, is the current of the jth winding and vI
is a coefficient corresponding to the jth winding which
field equations described above are respectively given
may for some values of j be a fractional number or
by zero. The RHS of eqn. 45 may then be expressed by
I I
6 H dl =
8 J j nJdaJ
3 (42)
and 3=1 J=1
where hc, = $vj The flux 4 is a function of the currents
B . nda = 0 (43) and MOsuch that $J = $J(il, ..., iJ, MO)and A,, = Ac$il,
These integral forms enable the circuit quantities h and ..., iJ, MO).Eqn. 45 is then expressed as
i to be deduced. Contour C is chosen so as to follow J
any single contour of flux, where H is related to B by
eqn. 35. Eqn. 43 will be written in the modified form
of By then summing the contributions from all the fila-
B . nBdaB = 4 (44) ments into which the field has been resolved yields
J
where da, may be any surface element of near infinites-
imal area which is orthogonal to an element of length w + W‘ = CXjij
dl and bisects C once and only once, as shown in j=1

198 IEE €’roc.-Electr. Power Appl., Vol. 145, No. 3, May 1998