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The dielectric properties of cryopreserved biological tissue are discussed in relation to the problems which arise when EM fields are used for rapid rewarming. The UHF band is favored in two ways: the avoidance of thermal runaway and the uniformity of heating inhomogeneous material. Various resonant cavity applicators are considered for efficient and uniform rewarming. The square-aspect TE 111 cylindrical applicator is favored principally because it allows variation of the E-field orientation as required during the warming profile. An appropriate kidney phantom organ is described. It is used to obtain measured values of the overall efficiency of a TE 111 applicator. The efficiency values are found to fall steadily with increasing temperature from 85% at -40 degrees C to 45% at the phase change, mainly due to the decreasing tan delta value of the phantom material.