The invention relates to a converter circuit (1), comprising a direct-voltage connection (13); at least one alternating-voltage connection (12); and at least one phase module (10), wherein the phase module (10) has a series connection of an upper and a lower converter valve (11a, 11b), wherein an electrical link between the upper and the lower converter valve (11a, 11b) is the alternating-voltage connection (12), wherein the connections (14a, 14b) of the phase module (10) are electrically connected to the direct-voltage connection (13), wherein the upper and the lower converter valve (11a, 11b) each have at least two sub-modules (20, 30), which are arranged electrically in series, wherein the sub-modules (20, 30) have at least one capacitor (42), at least one power semiconductor (41), and a control assembly (50) for controlling the power semiconductors (41), wherein at least one converter valve (11a, 11b) has a sub-module (30) suitable for linear operation, and wherein the converter circuit (1) has a control assembly (50), which enables the linear operation of the sub-module (30) suitable for linear operation. The invention further relates to a method for operating a converter circuit (1) according to the invention, wherein at least one sub-module (30) is operated in linear operation in at least one converter valve (11a, 11b).