We develop a semi-analytical theory for the self-bias formation in capacitively coupled plasmas excited by arbitrary radio-frequency (rf) waveforms. The requirement of rf current continuity and voltage balance across the discharge results in the need for a self-bias voltage to develop with non-sinusoidal excitations, even in geometrically symmetric systems. The theory is compared extensively with a wide range of experimental and particle-in-cell (PIC) simulation data within the literature, and is found to be in excellent agreement. Furthermore, it is shown that the present theory is formally equivalent to the original model proposed by Heil et al (2008 J. Phys. D: Appl. Phys. 41 165202), but goes further by explicitly allowing the time-varying sheath voltages and symmetry parameter to be evaluated without input from PIC simulations.