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2-threshold inverter (hardware, frequency response) - 11 Nov 2019

Here is a composite plot of the inputs, capacitor values, and outputs for periodic pulses from 8-26 ms (period = 16-52 ms, frequency = 62.5 down to 19.23 hz). Note that the output only switches for input pulses of 13 ms or greater (38.46 hz or less), thereby acting as a low-pass filter. For low and high output switching to occur, the input must be constant, high or low, long enough for the capacitor value to rise or fall from one threshold to the other: (Click for larger image)

The high and low input pulse widths are about the same. However, the output pulse widths are not equal. Here is a plot of average output pulse width (yellow = low, orange = high) with respect to average input high pulse width: The reason for the difference in output pulse widths is because the two threshold values (about 3.10 and 1.80 V) are not symmetric about the mid-supply value of 2.5 V (they are symmetric about 2.45 V). Therefore, it takes the capacitor a little longer to fall from the upper threshold to the lower one than it does for it to rise from the lower threshold to the upper one, and thus the low output pulse is correspondingly longer (by about 2.18 ms).

The single errant point in the plot above at 13 ms is due to the failure of the capacitor to fall completely to the lower threshold during one cycle of the input in a frame of logged data, and therefore the output misses generating a high pulse: The behavior of the output near to the low-pass cutoff should probably be examined in detail for anomalies such as this, as it is likely to be erratic, but not necessarily random.

Here is a 2d composite plot of the inputs and outputs for periodic pulse widths from 5-25 ms, varied both low (along the x axis), and high (down the y axis). Plots along the diagonal are similar to those above for equal high and low pulse widths: (Click for larger image)

Note that:

1. Along the diagonal, only signals below a maximum frequency (longer pulses) are passed.
2. Along the rows and colums, only signals with low and high pulse widths above some minimum value are passed.
3. In the third row and colum, only one signal is passed (on the diagonal). Signals with either low or high pulse widths longer than this value (but not both) are not passed. Here the device is acting as a band-pass filter.
4. Looking at the first row and colum, it is clear what is happening: as the low input pulses increase in width (first row), the capacitor values are pulled toward 0 V and do not cross the upper threshold, while as the high input pulses increase in width (first column), the capacitor values are pulled toward 5 V and do not cross the lower threshold. Only when the capacitor values stay near mid-supply, and span both the upper and lower thresholds of the device, will there be switched output.
5. An obvious question is whether the other (lower right) rows and colums are similarly constrained: i.e. as the low or the high pulse widths are increased away from the diagonal, does the output always eventually cease?

Here is another composite plot of periodic pulses from 5-50 ms (the upper left quadrant is similar to the above, but is new data): (Click for larger image)

In this plot:

1. The outputs in the fourth row and column also cease for increasing low and high input pulse widths, respectively. However, while the plots for high input pulse widths of 40+ ms show no output, a low pulse width of 50+ ms is required. This is again likely due to the slight asymmetry in the two device threshold levels.
2. As the low or high input pulses are increased further, eventually the capacitor will periodically saturate at either 0 or 5 V. However, in this case (e.g. at 0 V), the device can still be made to switch if a high pulse of at least 22 ms is provided (see the impulse response shown previously). Similarly, if the capacitor is saturated at 5 V, then a sufficiently long low pulse will cause the output to switch. Therefore, if both the low and high input pulses are at least as long as these values, the device will continue to switch output.
3. Overall, when the low and high inputs are equal, the device acts as a low-pass filter. When the low and high inputs are considered independently, the device acts first as a no-pass filter, then as a band-pass filter, and finally as a low-pass filter. Although the device has a single input and output, its behavior is 2-dimensional: a particular {low, high} input point is mapped to a particular {low, high} output point.

Here is a plot, from several angles, of the average input low pulse width (x axis), average input high pulse width (y axis), and the average output low and high pulse widths (z axis, yellow = low, orange = high) for each of the 10x10 runs (10 points per run) shown above. All axes are from 0-58 ms. Each of the output widths forms a discontinuous surface with respect to the 2d input. Both outputs are zero for points lying in the xy plane. The two surfaces intersect near the diagonal, although the yellow surface is slightly higher: The {low, high} pulse space can be interpreted as the real and imaginary parts of complex numbers (for non-negative pulse widths). Then, the action of the device is to perform a complex function on the input space, so that complex inputs are mapped to complex outputs. Here are plots of the input points from above (in yellow, low pulse width along x, high pulse width along y) and the corresponding output points (orange): Below, the two point sets are shown together. At right, the two point sets have been connected by lines to show how they are moved by the device: Note that:

1. Many points (outside the acceptance bands of the device) are mapped to zero.
2. Along the diagonal, points are moved slightly down and to the right.
3. The remaining points are reflected across the diagonal (since they swap low and high pulse widths), and are placed slightly farther away from it.

So, while this might not be a particularly interesting complex function (?), it is not a trivial one either.

Next: numerical responses.