Tracing Multi-valued Responses in Nonlinear Systems by Linking a Commercial HB Simulator to a Simplicial Algorithm

We present an auxiliary–generator (AG)–based outer–tier methodology that links existing simplicial path–following algorithms to commercial harmonic–balance (HB) simulators for the efficient tracing of multivalued solution curves in nonlinear circuits. The key idea is to formulate the AG nonperturbation conditions as a low–dimensional outer–tier system, while using the commercial HB engine as an inner–tier black box that evaluates the circuit response for each AG voltage setting. This AG–based variable reduction enables automatic determination of multivalued sections without manual parameter switching, optimization–based outer tiers, or access to internal HB variables, and it naturally extends to configurations with multiple AGs. On this reduced system, a simplicial path–following scheme traces the zero–error solution curve without Jacobian calculation or inversion, requiring only one HB solve per simplex. The proposed framework is demonstrated on two recently proposed systems for wireless power transfer.