Measurements of hysteresis losses W(h0, Hb) at several fixed amplitudes h0 versus a static bias magnetic field Hb collinear to h0 and directed along the length or the width of a VTi ribbon which exhibits significant magnetic anisotropy, are reported and are very well reproduced exploiting bulk pinning alone hence neglecting equilibrium diamagnetism and surface barriers. Families of curves of D≡W(h0, Hb)/W(h0,0) vs z≡Hb/h0 at various fixed h0 were calculated for planar and cylindrical geometry, neglecting surface steps, using the simple formulas μ0 jc=dB/dx=α/B, α/B0.5, α/B0.1, α[1−(B/Bc2)], and α/(B+Δ) for the bulk critical current density. For slab geometry and the first four functions listed, the curves of D vs z are independent of h0/H* when h0<H* (the first full penetration field) and z<zd with zd dependent on h0/H*. This decoupling of D from h0 occurs when xp, the maximum penetration of the flux disturbance during the cycle of the applied field, does not reach the midplane X. Closed form expressions for W(h0, Hb) developed for slab geometry using dB/dx=α/B confirm this by showing that under these circumstances h0 factors out of the formulas for D. Pursuing the condition xp=X, analytic expressions linking zd and h0/H* are obtained. Graphs of zmin, the locus of the valley minimum and of Dmin, the ratio of Wmin, the losses at the valley minimum to that at Hb=0 are presented. The edge of plateaus displayed by such graphs is determined by letting zmin=zd in the formulas for zd(h0/H*).

We have measured hysteresis losses, W (ho, hb' H bO ) ' in ribbons ofVTi and Nb subjected to oscillating magnetic fields of various amplitudes ho directed transverse to a static bias magnetic field H bO ' hence the name, noncollinear regime. A second bias field hb along ho may also be introduced. HbO and ho are directed along the length and width of the ribbon respectively or vice versa (90· rotation). The families of data curves are compared with the predictions of double criticalstate models where (i) dB I dx = ± F p (B )1 B governs the critical gradients of the magnetic induction and (ii) dO Idx = ± Fp(B )F(O )/B 2, the spatial variation of the orientation of the sheets offlux lines, with 0 = OalongH bo . The effect on W(ho,hb ,H bO ) ofchoosingF(O ) = l/tan 0, k and k I( 1 + 10 1)2 is illustrated and the latter is seen to yield the best agreement. The locus of the magnetization along the length and width of the samples, (M II ) and (M 1 ), have been monitored separately and simultaneously during the cycles of ho(t) for various ho andH bo with hb = ° or h o . These observations, in particular the curves of the magnetization along H bO ' constitute severe tests of any model. The corresponding curves generated by the model using F (0) = k I( 1 + 10 1)2 are presented. The physical origin of F(O) is explored.

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