Presently, the rf surface impedance Z of cuprate superconductors is still shrinking with material improvements, which is shown clearly by Z = Z1+Zres still dominated by extrinsic properties summarized in Zres. We present evidence that Zres is due to the large leakage current jbi and the small critical current jcj of weak links. The latter destroys the intrinsic shielding from a λ1 -thin seam λJ deep into the bulk. This causes rf residual losses Rres ≈ (ΩμO)2λJ3 σb1/2. Rres stays finite at T≃O by σb1(T→O)≈σb1(αjb1) being amplified by (λj/λ1>103 as a weighting factor. An appropriate measure of weak links is the grain boundary resistance Rbn(∝ρ(O)) enhancing λj ∝ Rbn and Rres ∝ Rbn2. Thus, Zres is minimal for minimal extrapolated normal conducting resistivity ρ(T→O).

To identify the weak links as new entity the H-field dependence is most helpful, because at very low fields Hc1J∝ 1/λJ Josephson fluxons penetrate into the weak links. These Josephson fluxons show negligible flux flow or flux creep, and enhance Zres by λJ(H, T) ∝ l/√Jjc (H, T). The measured JcJ (H, T) - and Jbl - values explain Zres quantitatively as well as in temperature ∝ ( a+ Tn) (n ≈ 1, T<Tc/2) and in field ∝ (b + Hn) (n≈1, H>Hc1J) dependence being thus a unique method to obtain the different Hc1-values. The strength of the field dependence d Zres/dH ∝ Zres (Hc1J*)/Hc2J(T) is not only a measure of Zres and HC2j(T) but is crucial for nonlinear effects and (fluxon) noise also, which limits the performance of rf devices.