Semi-grant-free (Semi-GF) non-orthogonal multiple access (NOMA) enables grant-free (GF) and grant-based (GB) users to share the same resource blocks, thereby balancing the connectivity and stability of communications. This letter analyzes ergodic rates of Semi-GF NOMA systems. First, this paper exploits a Semi-GF protocol, denoted as dynamic protocol, for selecting GF users into the occupied GB channels via the GB user's instantaneous received power. Under this protocol, the closed-form analytical and approximated expressions for ergodic rates are derived. The numerical results illustrate that the GF user (weak NOMA user) has a performance upper limit, while the ergodic rate of the GB user (strong NOMA user) increases linearly versus the transmit signal-to-noise ratio.