Experimental Investigation of Bifurcation Behavior in Nonlinear Microwave Circuits

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.

Coupling-induced hysteresis in free-running oscillators

“…The states S 1 and S 2 are stable, whereas U is unstable. Extending the techniques of [8] to the autonomous case, it is possible to stabilize the oscillator in state U. When the stabilization mechanism is turned off, the oscillator will jump (randomly) to one of the stable states S 1 or S 2 .…”

Analysis and Synthesis of Hysteresis Loops in an Oscillator Frequency Characteristic

“…can go up or down during the trace, but arc-length increases along the zero curve by construction. As described in the previous work [15], an application of an injection signal with nulled current does not change the state of the DUT, but can change its stability properties. In the case of an autonomous DUT, the frequency and phase of the oscillation of the DUT are determined by the reference oscillator (VCO) because the DUT injection-locks to it.…”

“…Y  can also be zero at two additional frequencies, below and above n  , as easily gathered from (15). Replacing the three frequencies in ( , ) r T Y V  one obtains three different amplitude values, one for each frequency, which explains the hysteresis induced by linear coupling effects.…”

“…A. Experimental apparatus The experimental results presented here have been obtained from an updated version of the technique described in [15]. Specifically:…”

“…The works [13]- [14] take a different approach using a contourintersection method, applied to an outer-tier admittance function extracted from HB. On the other hand, the work [15] addresses the experimental characterization of the full hysteresis loop in driven circuits. This requires the stabilization of section 2 [in the sketch of Fig.…”