Hysteretic ac loss of superconducting strips simultaneously exposed to ac transport current and phase-different ac magnetic field

A simple analytical expression is presented for hysteretic ac loss $Q$ of a superconducting strip simultaneously exposed to an ac transport current $I_0\cos\omega t$ and a phase-different ac magnetic field $H_0\cos(\omega t+\theta_0)$. On the basis of Bean's critical state model, we calculate $Q$ for small current amplitude $I_0\ll I_c$, for small magnetic field amplitude $H_0\ll I_c/2\pi a$, and for arbitrary phase difference $\theta_0$, where $I_c$ is the critical current and $2a$ is the width of the strip. The resulting expression for $Q=Q(I_0,H_0,\theta_0)$ is a simple biquadratic function of both $I_0$ and $H_0$, and $Q$ becomes maximum (minimum) when $\theta_0=0$ or $\pi$ ($\theta_0=\pi/2$).