  Purpose -- RF circuits often possess a multi-rate behavior. Slow changing
baseband signals and fast oscillating carrier signals often occur in the same
circuit. Frequency modulated signals pose a particular challenge.
Design/methodology/approach -- The ordinary circuit differential equations
(ODEs) are first rewritten by a system of (multi-rate) partial differential
equations (MPDEs) in order to decouple the different time scales. For an
efficient simulation we need an optimal choice of a frequency dependent
parameter. This is achieved by an additional smoothness condition. Finding --
By incorporating the smoothness condition into the discretization, we obtain a
nonlinear system of equations complemented by a minimization constraint. This
problem is solved by a modified Newton method, which needs only little extra
computational effort. The method is tested on a Phase Locked Loop with a
frequency modulated input signal.
  Originality/value -- A new optimal frequency sweep method was introduced,
which will permit a very efficient simulation of multi-rate circuits.
