Impressions vs Forms

We have seen that we can find fits that are very close to the actual characteristics for diodes. But fits bear the same relation to characteristics as an image does to an object. Our observations are not perfect and there are distortions present. The same relation exists between a print and the typeset page from which it was made. It is impression versus form.

The ancient Greek philosophers assigned a greater reality to the forms than our perceptions of them. The problem is similar to that of doing fits with the patterns being somewhat imperfect. But the same pattern may be applicable to a number of similar objects (in our case diodes) with different coefficients in the characteristic equations.

In The Sophist Plato uses an "Eleatic Stranger" with extraordinary skill to try to capture what we mean by a sophist. The Stranger first uses the dialect method in an attempt to pin down the sophist but he proves to be an elusive quarry. He appears to be many things at once. After studying "being" and "not-being" and what we mean by these terms he comes to the conclusion that the sophist is a pretender. He does not possess or impart true knowledge. The dialectic is a process of defining a pair of cases to which the object in question either belongs or does not in order to specify what it is. The Stranger looks down on the Sophist and is cutting him at the same time. The Sophist does not rank with the Philosopher.

The case is not quite so bad with the fits but one still can't trust them completely. Theory might help narrow down equations to be fitted but measurements usually have errors associated with them and can produce unavoidable residuals as a result.

5.1 V Zener Diode Conductance and Estimated Error in Voltages

We can refer to the ratio of small changes in current for small changes in voltage as a conductance*. It is the slope of the a line tangent to the V-I characteristics and in calculus is referred to as a derivative. The unit for conductance is the Siemens, S. If one plots this function for the 5.1 V Zener diode one sees that the conductance starts to increase rapidly at about 4.75 volts.



This conductance can be used to estimate the errors in the voltage measurements and compute corrections. The formula is ΔV = ΔI/G. The results for the steep part of the V-I characterists is shown below. This shows that even though the residuals are relatively large the corrections to the voltage needed to place the points on the curve are within about 1/2 the precision of the measurements and appear to be just round off errors.

The residuals for these points were more randomly distributed. The "corrections" for lower voltages were larger where the curve was flatter and where the fit appeared to deviate somewhat from the data. The conclusion is that relatively larger residuals are to be expected where the slope of the curve is relatively steep.

*edit: This is not the usual conductance which is defined in terms of Ohm's Law, G = I/V. My terminology is somewhat lax. A more appropriate expression might be "small signal conductance" or "differential conductance."

Does Least Squares Make the Best Choice for Coefficients?

Least Squares is often used to find a fit to a set of data. Linear Least Squares is easier since the solution for the unknown coefficients only involves a set of equations which are linear functions of the unknowns. The primary condition for a least squares fit is that the sum of the squares of the residuals be a minimum. The result is that portions of the fitted curve can have a definite deviation from the data set and still result in a minimum value for the sum.

For the fits of the data for the Zener diode I used Mathcad's genfit function. For the composite functions of the last fit the individual fits can also be done by fitting lines in a log-log plot*. One of the problems that I encountered in doing the fits is that the fitted function tended to cut corners in the region of the "knee" where the characteristics change from lower currents to higher ones. As a result there is a correlation among neighboring residuals there. Intuitively one would expect that the residuals would be randomly distributed with respect to the fitted curve for the best fit. There seemed to be something wrong with the residuals.

I decided to give the rms residual more control in doing the fit. I tried dividing the data into sections with each section being represented by a term in the fit function. The result is that the residuals differed for each region. This method also seemed somewhat arbitrary since the terms for the function depended on how the data was partitioned. The results suggested the use of a Taylor series for the current involving exponential functions. Fitting a quadratic function of the first fit didn't work. The difference between the fit for lower currents and the data looked closer to an exponential function that was fairly linear for lower currents so I tried multiplying the current by a factor involving an exponential term. That worked much better than the Taylor series. Varying the number of data points included in the first fit optimized the final fit. The rms residual was closer to the result of a tedious process involving a search for the coefficients in a region of fixed size and a random choice from among a few of the lowest values of the sum of the squares of the residuals to avoid getting "stuck in a loop." (The results of the second fit method weren't consistent. The coefficients posted were the best values found.)

The results show the utility of the residuals in analyzing the results of a fit and comparing one fit method with another.

The economists like to use linear models for optimization problems but nonlinear models may prove to be more useful. Fits can be used to find approximate functions for observable relations in a theory. Doing what is best for the individual is not necessisarily at odds with doing what is best for the group. The problem may be similar to that of finding the best fit for a set of data.

*edit: One would have to do a semi-log plot. For the first fit involving the lower curve one would also have to ignore the small constant, u_0, and plot log(I1/mA) vs V. For the second plot it would be log(I2/I1 - 1) vs I1 (I2>I1) with I1 being the current in the first fit and I2 that of the second fit.

Impossible Choices

Arrow's Social Choice and Individual Values and Sen's Liberal Paradox point out some of the failings of the democratic process and the impossibility of reaching a group concensus on a number of options. But this just validates the need for a mechanism to make a decision in the absence of a majority or the situation where the outcome is unclear. In the case of a tie in the US Senate the Vice President gets the deciding vote. Congress is empowered to decide the outcome of the elections of its members. In these situations the decision that is made is not democratic since the voters failed to reach a decision.

If the choice is left solely to individuals though some people can end up benefiting at the expense of others. The most powerful factions end up being self-serving and disregard the interests of others.

Arrow also points out that capital is not the only way to improve productivity. Technological innovation can substitute for capital. This includes the arrangement of the workplace. Improved economy has its benefits. There are other ways of improving. One is education which improves the value of workers. And an economic slowdown may indicate that there are forces at work which tend to obstruct progress.

The Problems With Fitting Zener Diode Characteristics

Part of the difficulty in fitting the V-I characteristics of a Zener diode seems to be due to the current being a exponential function of an exponential function resulting in a very rapid change in the conductance of the diode. If one fits the characteristics for smaller currents one gets an exponential function for the current. The steep portion of the curve appears to be an exponential function of the first current. As a result the conductance, G, of the diode—the slope of the curve at a particular point—becomes quite large. At any point the change in current ΔI = G ΔV, the conductance times the change in voltage so a small error in measuring the voltage will result in a large shift in the current measurement.

The fit above consisted of two fits, the first for current of the lower part of the curve and a second fit to determing the diode current as a function of the first current. Both curves are easy to fit. The rms residual is still large compared to the precision of the measurements which was 0.01 mA for the current and the 0.01 volts for the voltage. In the plot of the residuals below one sees large deviations at the higher currents. The relative deviations (residual/current) are of the same magnitude for all currents

The current being a function of a current is consistent with a controlled breakdown. There is also a certain amount of noise present and the current fluctuates some due to a statistical component. The largest contribution to the rms deviation is from the upper portion of the curve.

Supplemental: The correlation among neighbouring residuals near I = 0 mA indicates that the fitted functions are not quite right but this error is small compared to larger residuals which may have contributed to errors in the values of u and v.

A Better Fit for the 5.1 V Zener Diode

By using the diode equation and tinkering with the fit procedure I was able to get a better fit for the 5.1 V Zener diode.

I made another set of measurements and they are in good agreement with the first set. There still seems to be some residual error greater than the precision of the measurements. Most of the error is above 4.0 volts. So I should probably checked for internal resistance.

RadioShack 5.1 V Zener Diode V-I Characteristics

Over the weekend I've been studying the characteristics of a RadioShack 5.1 V Zener Diode. The Zener diode is used as a voltage reference since its relative change in voltage is small compare to a relative change in current above the indicated voltage. I was able to get a good fit to the measured characteristics**.

The form of the equation that I used to fit the characteristics curve just involved variables in the exponents. It is by no way unique as can by the fact that a multiplying current can be canceled out by a constant in an exponent. The fit proved challenging since my computer kept encountering singularities and it was difficult to find a starting point for a solution.

I was able to get a fairly good fit by fitting the upper and lower branches of the curve separately. The lower part is just the regular current for a reversed diode and an exponential curve is a good approximation. The upper branch includes in addition the breakdown current for the diode so one can fit the difference between the measured values and the extrapolated fit for the lower part of the curve. I also tried a linear estimate of the zero* of the variance but that proved rather slow. A quadratic estimate of the zero of the variance proved to be quicker after one got away from β = 0 where there was a singularity.

The fit shown started with a linear estimate of the zero and then switched to a quadratic estimate for faster convergence. It confirms the presence of two currents.

*edit: It was the zero of the least squares fit for which the variance is a minimum.

**edit: I corrected for the voltage drop across the milliammeter and got a steeper slope above 5.1 V and the rms deviation, δ'_rms, reduced to 0.320 mA which probably means that there is better agreement with the assumption of two currents. The diode may also have some internal resistance.

NIO Sets His Sights the Moon

NIO did a little more stargazing this morning and managed to capture an image of the Moon.

I found that there were more manual exposure options available but it was difficult to get the focus right with the optical zoom since the image wasn't in focus with the focus set at ∞. I'm going to have to work on that. Using as wide an aperature as possible would probaby help some but without a telescope I doubt if it will be possible to capture much of the fine detail.