Abstract We show how the hysteresis behaviour in electronic circuits can be explained in a robust manner using PSpice transient simulations. Furthermore, we describe a simple circuit for three-segment nonlinear characteristic shaping and show how this circuit can be used to produce hysteresis loops. The realisation of relaxation oscillators is then given as a typical application.

Keywords amplifiers; hysteresis; nonlinear circuits; nonlinear dynamics

(ProQuest Information and Learning: … denotes formulae omitted.)

Undergraduate electronic circuits textbooks, such as Ref. 1, introduce the hysteresis phenomena to their readers while attempting to explain the behaviour of a simple op-amp comparator with positive feedback. Such a circuit is then characterized as bi-stable; i.e. it has two different driving point characteristics depending on whether the input voltage is increasing or decreasing. When plotted together on the same V^sub i^-V^sub o^ chart, a discontinuous loop appears and is termed the hysteresis loop. The sudden jumps in the loop are attributed to the bi-stable nature of the circuit and are not discussed any further.

From a nonlinear dynamics point of view, sudden jumps correspond to very fast energy transfer. Accordingly, an energy storage element (capacitor or inductor) of a significantly small value must exist to hold this transit energy transfer. Including a parasitic capacitor (inductor) is necessary but not sufficient to explain the hysteresis jumps, as explained in detail in the pioneering work of Kennedy and Chua2 which ironically never found its way to any textbook. In particular, the two necessary conditions for hysteresis to occur are:

1 Astatic d.c. nonlinear non-monotone driving-point characteristic.

2 Aparasitic energy storage element to accommodate the fast energy transfer. This fast transition is described by a stiff differential equation (one which contains two widely separated time constants).

The work presented here has two objectives:

(i) To show how the hysteresis behaviour can be explained in a clear manner with a circuit simulation example.

(ii) To show how nonlinear driving-point characteristics can be designed.

Explaining hysteresis

To explain hysteresis, one should address the two necessary conditions stated above. The first condition relates to the existence of d.c. nonlinear characteristics. In Fig. 1, the four basic nonlinear characteristics which can be obtained from any three segments are shown. The characteristics of Figs 1(a) and 1(b) are monotone which means they are equally controllable via the x-axis or the y-axis variable. A single value for x corresponds to a single value for y. However, Figs 1(c) and 1(d) are non-monotone because they are controllable only through the y-axis variable. Attempting to control these characteristics through the x-axis variable results in multiple values of y for the same value of x, which is not possible in electronic circuits. Accordingly, the apparent characteristics acquire ‘discontinuous jumps’ and follow the dashed lines in Fig. 1. Stimulating hysteresis is thus associated with controlling a nonlinear non-monotone driving-point characteristic via the wrong control variable.