Many displacement transducers exhibit hysteresis in their response to drive signals. To observe this behavior, drive the actuator with a triangle wave while measuring the actuator response with a linear position transducer. Then, display the command and response signals on an X/Y oscilloscope. Figure 1a depicts the familiar lenticular hysteresis display at a low-repetition frequency.
Typically, an actuator with such response exhibits more hysteresis as the triangle-wave frequency is increased. If we could superimpose the trace at a higher frequency over the low-frequency trace, we would see a display such as that depicted in Figure 1b.
Since this actuator response to a triangle-wave input can be measured in advance, a nonlinear drive signal can be generated to compensate for the response. This is achieved by making an amplifier that has approximately the opposite re-sponse to the triangle-wave input. Driving the actuator with the properly warped signal makes the position-displacement response more linear with respect to the triangle-wave input signal. For the triangle-wave input signal, VIN, the desired drive signal, VOUT , is determined as shown in Figure 2 (dashed line).
The required signal has a different response to rising input voltage than it does to falling input voltage. This suggests a straightforward circuit implementation. If a delayed version of the input signal is subtracted from the present value of the input signal, a signal VUP/DOWN can be established that is proportional to the input dv/dt. With a sample/hold amplifier and a difference amp, a ΔV signal can be produced. Figure 3 depicts an appropriately scaled version of VUP/DOWN for a triangle-wave drive signal. Adding a scaled and low-pass-filtered output of VUP/DOWN to the original triangle wave produces a warped drive signal (Fig. 4).
Operation of the sampler at a fixed clock rate means that as the frequency of the example triangle wave increases (or as the slew rate of any drive signal increases), so too does the amplitude of VUP/DOWN. Therefore, the warping increases with frequency in a manner appropriate to compensate for the increasing actuator nonlinearity.
For the first breadboard version of the pre-warper, a simple first-order lowpass filter was selected for the VUP/DOWN filter. The breadboard schematic can be seen in Figure 5.