VxV×A+σ V.A + uo where o has been assumed piecewise constant. This choice of gauge decouples the vector potential from the scalar potential. For the important class of two-dimensional problems there will be only one component of A parallel to the excitation current density. For fields involving time, at least two types can be distinguished: the time harmonic(ac) case in which the fields are periodic at a given frequency o, i. e, A=A. exp(jot), and the transient case in which the time dependence is arbitrary Low Frequency Fields In the important class of problems belonging to the low frequency limit, i. e, eddy current effects at power frequencies, the second derivative terms with respect to time(wave terms)in Eq.(44. 1)vanish. This approxi- mation is valid if the dimensions of the material regions are small compared with the wavelength of the prescribed fields. In such circumstances the displacement current term in Maxwells equations is small compared to the free current density and there will be no radiation [Stratton, 1941. In this case, while a full vector field solution is necessary in the conducting regions, in free space regions, where o =0 and curl H=Js Eqs. (44. 1)can be eplaced by V2i=0, where t is a scalar potential defined by H=-vd. The scalar and vector field regions are oupled together by the standard interface conditions of continuity of normal flux (b)and tangential field(H) Statics limit In the statics limit(dc) the time-dependent terms in Eq (44. 1)vanish, and the field can be described entirely by the poisson equation in terms of a single component scalar potential, which will be economic from the numerical point of view. In this case the defining equation is V·μVd=V·pH (44.2) where d is known as the reduced magnetic scalar potential with H= H,-va, and H, the source field given by the biot Savart law. Some needed in solving Eq.(44.2)numerically, in practice, as H, will often be calculated to a higher accuracy than For instance, in regions with high permeability(e.g, ferromagnetic materials), the total field intensity H tends to a small quantity which can lead to significant errors due to cancellation between grad and H,, depending upon how the computations are carried out. One approach that avoids this difficulty is to use the total scalar potential y in regions that have zero current density [Simkin and Trowbridge, 1979, i.e., where H=-Vo and H is the coercive field for the material ap satisfies v·μψ=V·μH Again, the two regions are coupled together by the standard interface condition that results, in this case, in a potential" jump"obtained by integrating the tangential continuity condition, i.e φ=y+|Hdr (444) where I is the contour delineating the two regions that must not intersect a current-carrying region; otherwise the definition of v will be violated c 2000 by CRC Press LLC© 2000 by CRC Press LLC (44.1) where s has been assumed piecewise constant. This choice of gauge decouples the vector potential from the scalar potential. For the important class of two-dimensional problems there will be only one component of A parallel to the excitation current density. For fields involving time, at least two types can be distinguished: the time harmonic (ac) case in which the fields are periodic at a given frequency v, i.e., A = Ao exp(jvt), and the transient case in which the time dependence is arbitrary. Low Frequency Fields In the important class of problems belonging to the low frequency limit, i.e., eddy current effects at power frequencies, the second derivative terms with respect to time (wave terms) in Eq. (44.1) vanish. This approxi￾mation is valid if the dimensions of the material regions are small compared with the wavelength of the prescribed fields. In such circumstances the displacement current term in Maxwell’s equations is small compared to the free current density and there will be no radiation [Stratton, 1941]. In this case, while a full vector field solution is necessary in the conducting regions, in free space regions, where s = 0 and curl H = Js Eqs. (44.1) can be replaced by ¹2 c = 0, where c is a scalar potential defined by H = –¹c. The scalar and vector field regions are coupled together by the standard interface conditions of continuity of normal flux (B) and tangential field (H). Statics Limit In the statics limit (dc) the time-dependent terms in Eq. (44.1) vanish, and the field can be described entirely by the Poisson equation in terms of a single component scalar potential, which will be economic from the numerical point of view. In this case the defining equation is ¹ • m¹f 5 ¹ • mHs (44.2) where f is known as the reduced magnetic scalar potential with H = Hs – ¹f, and Hs the source field given by the Biot Savart law. Some care is needed in solving Eq. (44.2) numerically, in practice, as Hs will often be calculated to a higher accuracy than f. For instance, in regions with high permeability (e.g., ferromagnetic materials), the total field intensity H tends to a small quantity which can lead to significant errors due to cancellation between grad f and Hs , depending upon how the computations are carried out. One approach that avoids this difficulty is to use the total scalar potential c in regions that have zero current density [Simkin and Trowbridge, 1979], i.e., where H = –¹c and Hc is the coercive field for the material c satisfies ¹ • m¹c 5 ¹ • mHc (44.3) Again, the two regions are coupled together by the standard interface condition that results, in this case, in a potential “jump” obtained by integrating the tangential continuity condition, i.e., (44.4) where G is the contour delineating the two regions that must not intersect a current-carrying region; otherwise the definition of c will be violated. —¥ —¥ + + = — —× È Î Í Í ˘ ˚ ˙ ˙ + = —×— 1 1 2 2 2 2 m s ¶ ¶ ¶ ¶ m m ¶ ¶ ms ¶ ¶ A A A A t t V t V t V e e f y = + Ú Hs dG G 0