SPICE Hysteresis Modeling Solution

This article presents a technique for constructing a standard SPICE model that is able to model the essential characteristics of systems with continuous hysteresis.

The example model in this article shows adding a high-frequency sine wave offset to an audio signal to reduce signal distortion on an analog tape recorder. This technique allows for a sufficiently accurate approximation to illustrate the distortion created by hysteresis and how it can be reduced by adding a high-frequency offset.

Hysteresis Model

The essential problem with modeling hysteresis is that it is a static or DC effect with memory. That is, the next value depends not only on the current value, but also on the previous value. However, this last value dependency does not depend on time. This results in a multi-valued transfer function.

Unfortunately, standard SPICE does not directly support this type of modeling. All reliance on the last value in SPICE is usually the result of a linear integration, which inherently leads to a frequency-dependent transfer function and does not take into account the distortion mechanisms.

One way to solve this problem is to simply realize that one can cheat. Analog models only need to do approximately what they need to do over a limited frequency range. Analysis shows that a small capacitor combined with a nonlinear diode resistance can be used to continuously store the last value of the signal before it changed slope direction to provide effective hysteresis, but without excessive dependence on frequency.

This is in contrast to some SPICE "hysteresis" models which only have two output state models and do not allow for a continuous transfer function.

Linear Model

The following schematic forms the basis of a continuous hysteresis model that can be used to model; for example, a magnetic core.

Note that the output voltage here is multi-valued but essentially linear beyond the dead zone. The dead zone occurs when the signal changes direction. It can be adjusted by the diode parameter N.

Figure 1. Schematic diagram of the continuous lag model

The output voltage of this module essentially follows the input linearly, but with an offset voltage. When the input is reversed, the capacitor holds the voltage so that there is a dead band from the peak voltage reached.

The key principle of operation is the presence of a nonlinear impedance that has a sharp resistance ratio under forward and reverse bias conditions. The standard diode equation is the simplest, but not essential, equation for this technique. It is used here to illustrate the method.

Substitution equations can be used to fine tune the response characteristics. The input voltage can also be further processed to obtain different nonlinear transfer curves. The example here uses the behavioral model of a diode:

b1 aci={is}*（exp（{k}*v（a，c）） - 1）

To obtain an accurate model, the values ​​of the components are chosen so that the frequency effects are minimized over the frequency range in which you wish to model the system.

The time constants of R load and C memory are such that the last voltage before the turn around does not leak too much. The charging current through the driven impedance (i.e. the diode in this particular case) does not limit the response of the system over the desired operating frequency range.

The above topology produces the following set of transfer functions and hysteresis plots for various input voltages and frequencies:

Figure 2. Ramp Input Transfer Function - F=1KHz, VIN=2V, 4V, 6V, 8V, 10V

Figure 3. Ramp Input Transfer Function - F = 1MHz, VIN = 2V, 4V, 6V, 8V, 10V

Figure 4. Hysteresis - F=1kHz, VIN=2V, 4V, 6V, 8V, 10V

Figure 5. Hysteresis - F=1MHz, VIN=2V, 4V, 6V, 8V, 10V

The key point of the graph is that over a frequency range of 1000:1, the voltage transfer function and hysteresis voltage are relatively constant, thus providing a good approximation of true DC hysteresis.

Typically, one constructs a SPICE behavioral resistor from a controlled current source with the desired forward and reverse characteristics. For example, as we pointed out above, the hysteresis deadband voltage can be adjusted by changing the diode parameter “N” from its default value of “1”.