Electronic circuits can be described mathematically by a system of (nonlinear) differential algebraic equations (DAEs). In case of autonomous oscillators the frequency, at which the oscillator is in steady-state, is an additional a priori unknown leading to an under-determined system of equations when employing Harmonic Balance. Standard approaches for gauging the solution are solving the resulting system by a Moore-Penrose pseudoinverse or by fixing the phase of one harmonic. In both cases Newton-like techniques show poor convergence conditions in practice. By the introduction of an additional periodic probe voltage source to one node of the oscillator circuit we can reformulate the autonomous DAE with the unknown periodic steady state as a non-autonomous DAE with the additional constraint that the current through the source has to be zero for the limit cycle. In a two stage approach we solve in a first step the system of equations keeping amplitude of the additional source and frequency fixed until convergence of Newton's method. In the next step the amplitude and the frequency will be updated by solving the additional constraint. The algorithm proceeds with the first step until convergence of the outer loop is achieved. The under-determined system of the additional constraint can be solved either by a Moore-Penrose pseudoinverse or by fixing the phase of one harmonic of the stimulus. If only one Newton iteration will be performed at the inner stage, we get a fast one stage approach. In practice additional techniques are necessary to improve robustness. Application of deflation techniques (divide system of equations by Euclidian norm of probe amplitude) prevents the algorithms from tending to the trivial solution. The small range of convergence towards the limit cycle for the initial probe amplitude can be expanded drastically by employing the affine invariance technique as damping strategy to Newton's method. The affine invariance technique shows superior properties compared with (modified) steepest descent techniques and homotopy methods requiring only slight additional overhead. Applying the affine invariance technique to the one stage approach leads to a fast algorithm for limit cycle calculations with tremendously improved range of convergence. The more robust, but slower alternative is the two stage approach in conjunction with the affine invariance technique.

* University of Bremen, Institute for Electromagnetic Theory and Microelectronics - ITEM, P.O. Box 330440, D-28334 Bremen / Germany
** Philips Research Laboratories, ED&T/Analogue Simulation, Prof. Holstlaan 4, 5656 AA Eindhoven / The Netherlands.