Real and causal hysteresis elements

Hysteresis is a phenomenon that has been observed across many different materials and situations. Under small-amplitude cyclical motion, classical hysteresis designates a constant loss per cycle over a wide range of frequencies. This is also consistent with an increase in losses or attenuation with frequency that is strictly proportional to the first power of frequency. Unfortunately, the classical (and simple) frequency domain description of hysteresis does not result in a real and causal impulse response, and therefore is not useful for predicting laboratory results. This problem has led to many errors as well as other more fruitful approaches over the years. The frequency domain requirements for hysteresis are re-examined and it is demonstrated that there is a family of solutions that provide real and causal impulse responses over some extended frequency range. The family is conveniently divided into highpass, lowpass, and bandpass causal systems. These are populated by closed form analytical solutions which can be applied to the prediction of motion and waves in hysteretic materials and systems.