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Radio-frequency 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. The paper aims to discuss these issues.

The ordinary circuit differential equations are first rewritten by a system of (multi-rate) partial differential equations in order to decouple the different time scales. For an efficient simulation the paper needs an optimal choice of a frequency-dependent parameter. This is achieved by an additional smoothness condition.

By incorporating the smoothness condition into the discretization, the paper obtains a non-linear 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.

A new optimal frequency sweep method was introduced, which will permit a very efficient simulation of multi-rate circuits.