In order to fulfill the grid codes and meet power quality requirements in grid-connected converters without increasing the switching frequency or system size, higher order line filters are often used. This paper considers the active damping of $LC\!L$-filters in grid-connected converters. Analysis in the discrete time domain shows that the damping can be increased when proportional plus derivative control of the capacitor current is applied, and it is found that this scheme has a simple implementation in a discrete time system as it is equivalent to proportional feedback of the last two samples. To find optimal damping controller gains, an analytical solution is derived for the open-loop system, and for the closed-loop system, a numerical optimization algorithm is developed. The performance of the proposed methods is compared using analysis in the $z$-domain, simulations, and experimental investigation. The analytical solution has good damping, an easy implementation, and is seen to have robustness against parameter variation, whereas the numerical solution is computation intensive but has even higher damping.