The resonance frequency in ultrasonic transducers is affected by the amplitude of power or current delivered. In the figure above, an LCR circuit's admittance is plotted for a linear LCR circuit and a non-linear LCR circuit. In the non-linear circuit, the capacitance is a function of current.

As the current increases, the capacitance increases. The result of this effect is that the resonance frequency decreases and a sharp transition is seen at the new resonance frequency.

For the plot above, the LCR values were:

• R=1110 Ohm, L =0.19 H, and C= 15.1nF.

In the nonlinear resonance, the R and L were the same, but the capacitance was defined as:

• [15.1 nF]*(1+current/0.007)^2

By using small frequency step sizes, I did not have to get fancy with the determination of current, as I could just use the current at the previous frequency step.

This representation of capacitance results in a similar maximum current amplitude to the linear model, but gives the distinct nonlinear transition and resonance frequency shift seen when doing a constant voltage sweep over the resonance frequency of an ultrasonic transducer.

For small signal excitation, the results are linear. However real-world ultrasonic transducers do not operate with "small signals", so finding appropriate analysis methods is critical for a deep analysis.