Harmonic balance and almost periodic inputs

We consider the equations of a large class of nonlinear circuits driven by asymptotically almost periodic inputs, and give an analytical basis for the use of harmonic balance to find steady-state solutions. More specifically, we show that in a certain setting of general interest there is a unique solution to the problem of obtaining a harmonic balance approximation, and that the approximations approach the actual solution as additional spectral components are included. Since any finite sum of sinusoidal functions with arbitrary frequencies is an almost periodic function, the results are of importance in connection with e.g., the determination of intermodulation effects. Our results involve a key circle-condition hypothesis. We give an example which shows that this hypothesis cannot be relaxed significantly. The example is of interest also with regard to the stability of feedback systems.