The analytical solution for the sub-module voltage ripple equations of a Modular Multilevel Converter (MMC) is derived, based on the knowledge of the external voltage/current magnitudes and enhancing a concept previously presented in literature. In order to achieve high accuracy, all passive elements of the converter, common-mode voltage injection as well as intentionally imposed circulating current harmonics are taken into consideration. A three-phase grid-connected MMC is chosen as an application example. The line and circulating current dynamics are also derived, which show that they can be controlled independently by forming two respective feedback loops. Any additional voltage regulators are avoided. The proposed implementation utilizes proportional-resonant (PR) terms for the line current, making it thus possible to introduce explicitly any chosen harmonic component. Experimental results from a reduced-scale laboratory prototype verify the discussed concepts.