Abstract
We present an experimental technique to study bifurcations in periodically forced nonlinear microwave circuits, including even physically unstable (periodic) steady states. The designer specifies a key node in the circuit being studied (often associated with an active device) and the method synthesizes a voltage waveform to match the waveform at the selected node so that no current flows across the interface. This null condition is maintained while a parameter, such as bias voltage, is varied over a specified range. The addition of the external nulling source is able to stabilize a steady state that would be unstable in the original circuit. Various applications are presented.
Highlights
N ONLINEAR microwave circuits under periodic excitation can exhibit a variety of non-standard behaviors including [1]–[3]: multiple steady-states, hysteresis and jump phenomena with respect to variation of a parameter [4], [5], sub-harmonic solutions [6], parametric oscillations, quasi-periodic solutions [7], and chaotic solutions [8]
We assume the reader is already familiar with a steady-state simulation method such as harmonic balance (HB) [3], [13] in which each periodic steady-state waveform of a circuit is captured with n real numbers (e.g., Fourier coefficients)
The independent variable in the graphic is the bias voltage applied to the varactor and the dependent variable is the magnitude of the fundamental Fourier term of the voltage waveform across the varactor
Summary
N ONLINEAR microwave circuits under periodic excitation can exhibit a variety of non-standard behaviors including [1]–[3]: multiple steady-states (typically with different stability properties), hysteresis and jump phenomena with respect to variation of a parameter (such as bias voltage) [4], [5], sub-harmonic solutions [6], parametric oscillations, quasi-periodic solutions [7], and chaotic solutions [8]. If a circuit based on a new semiconductor device exhibits some type of bifurcation behavior before a numerical model for it is available, simulation techniques cannot be applied. In a more extreme case, the provider of a system-level component might consider detailed modeling data proprietary and not want to reveal them Bifurcation behavior of such designs can only be investigated experimentally. In other cases (in particular, power amplifiers [4], [5] or some recent MEMS devices [19]), hysteresis is undesirable and the method can be used to confirm that it has been eliminated Other circuits such as microwave frequency dividers have inherently different regions of operation, delimited by bifurcation phenomena.
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