The Immersion and Invariance (I&I) wind speed estimator is a powerful and widely-used technique to estimate the rotor effective wind speed on horizontal axis wind turbines. Anyway, its global convergence proof is rather cumbersome, which hinders the extension of the method and proof to time-delayed and/or uncertain systems. In this letter, we illustrate that the circle criterion can be used as an alternative method to prove the global convergence of the I\&I estimator. This also opens up the inclusion of time-delays and uncertainties. First, we demonstrate that the I&I wind speed estimator is equivalent to a torque balance estimator with a proportional correction term. As the nonlinearity in the estimator is sector bounded, the well-known circle criterion is applied to the estimator to guarantee its global convergence for time-delayed systems. By looking at the theoretical framework from this new perspective, this letter further proposes the addition of an integrator to the correction term to improve the estimator performance. Case studies show that the proposed estimator with an additional integral correction term is effective at wind speed estimation. Furthermore, its global convergence can be guaranteed by the circle criterion for time-delayed systems.