Closed-Form Formulas for Predicting the Nonlinear Behavior of All-Pole Bandpass Filters

The nonlinear behavior of the all-pole passive bandpass filters close to their center frequency is studied using first-order approximations. The focus of this study is on approximate calculation of third-order intermodulations through an iterative process, and it is shown that it can be generalized to higher order effects. Nonlinearities in both resonators and inverters are considered. Simple formulas are given to calculate the effect of individual elements in the overall response in lossless and lossy filters. It is shown that the presence of loss will increase the third-order intermodulation intercept point of the filter, while it usually decreases the output third-order intercept point. In comparison to the nonlinear resonators, the relative effect of nonlinear inverters is larger; however, the losses of the inverters are found to have a smaller effect on the nonlinear response. Finally, simple formulas are given to predict the changes in the center frequency due to the reactive effects of the nonlinearities. In all cases, the analytical results are verified through comparison with harmonic balance simulations.