  A new self-consistent semi-analytical method for calculating the stationary
beam-induced voltage in the presence of arbitrary filling patterns and
impedance sources in storage rings is presented. The theory was developed in
space-domain with resonator wake-functions and in frequency-domain with
arbitrary impedance functions. The SIRIUS storage ring parameters were used to
benchmark the results, demonstrating good agreement between the two approaches
and with macroparticle tracking simulations. Additionally, a different approach
to simulate the beam-loading compensation of active rf cavities was
investigated in frequency-domain, proving to be a more generic description than
the methods generally used. The impact of broadband impedance on the
longitudinal equilibrium was straightforwardly evaluated with the
frequency-domain framework, without intermediate steps such as fitting
broadband resonators or convolving short-range wakes with bunch distributions.
Finally, a simple study of Touschek lifetime improvement with a passive higher
harmonic cavity is presented.
