I do not usually write lab notes on specific programming issues. I try to keep my web site general so many people can understand it. This note is an exception, its a real simple software related issue. Over the years of writing software I keep coming back to DeMorgan’s Theorem. It was one of those things that in college I never thought I would use, but it seems to pop up on a regular basis at work. I think this is fascinating stuff, particularly when academia meets the pavement.

First, the background. DeMorgan’s Theorem states:
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DeMorgan’s theorem cannot be proved or reduced using math. The only proof is a brute force proof. At this point your probably thinking, "how can this academic theorem prove to be useful at work?" Let me show by example.

This is going to be a real simple contrived example to demonstrate DeMorgan. Suppose the programming job is to read two switch inputs and to turn a motor on if both switches are closed. Perhaps to drive the motor ON we also have to output a LOW. It is common in computer to have negative I/O logic. Here is a schematic for this. We can see that if either switch is closed the input to the computer will go low.

I can now write software to implement the required function.

If ( SW1 = 0) AND ( SW1=0)
THEN MOTOR = 0
ELSE MOTOR =1

Do you see the correlation between this software statement and DeMorgan’s Theorem? Now we can simplify the programming statement using DeMorgan’s. Two simplifications make the statement easier to read. We take advantage of BOOLEAN type's and we can switch around the THEN and ELSE part to get rid of a NOT operator.

If (SW1 OR SW2)
THEN MOTOR =1
ELSE MOTOR =0

An astute programmer might note a further simplification by writing a single algebraic Boolean expression for this statement such as

MOTOR = SW1 OR SW2.

Understanding and using DeMorgan’s Theorem has simplified the programming, if only a little here. In larger applications we would get a larger return on our simplification investment.

Even though many compilers will reduce expressions internally, simplifying your programming pays rewards in understandable code. ALWAYS simplify code and remember to use DeMorgan.

I am staying at an apartment in China for a few weeks.  The other day the circuit breaker to my air conditioner popped.  I knew I smelt a funny smell coming from the breaker panel!  It finally went altogether last night.  Here is a picture of the breaker that the electrician pulled from the panel.  When I saw this circuit breaker I knew I had a good visual and needed to write up a lab note for my web site.  I saw the electrician do something smart after he pulled the breaker, he went and tightened up the screw terminal's for all the other breakers.

So whats the story surrounding aluminum wiring?
Many houses that were built in the late 1960's and early 1970's in the US were built with aluminum wiring.  Aluminum wiring was used because copper starting becoming very expensive around 1965.  Copper is now so expensive that my friend Alan Triolo, who is an electrician, makes some significant pocket change by going to renovation site's, ripping out the old copper wiring and selling it (with permission of course).  Another story comes to mind.  A few years ago the state of Nevada was having a serious vandalism problem.  The street lighting in some of desert locations was not working.  It seems vandals were ripping out the copper wiring from the street lighting to sell.

One final story.  When I was younger my family moved from Brooklyn to a nice, new, suburban home.  That house had aluminum wiring.  It was not until the 1970's that the issues surrounding aluminum wiring became well known.  When we found we had aluminum wiring we lived with a lot of anxiety.  Let me first try to calm you if your house has aluminum wiring.  I have not been able to find know cases of fires caused by aluminum wiring.  The few real problems seem to be related to incorrect installations.  On the other hand there are real issues to be concerned with.  You may not be able to get insurance unless this issue is taken care of.  The potential for a real problem is very real as my recent experience shows.

You do not need to rip out all the wiring in your home, though that is a possible solution.  Get a licensed electrician and talk over the alternatives.  He needs to go through the electrical devices in your home and make sure that they are rated to connect to Aluminum wire at a minimum.  Better yet, he can pigtail the ends of the aluminum wire with copper wire.  And finally he should tighten any loose or suspicious terminal connections on all Aluminum rated devices.

A couple months ago I ran across a circuitry problem that may interest those with a little electronics knowledge. We were getting failures on a high percentage of new product, but not on all of them. The product was warning that its internal -15v supply was out of tolerance (and faulting). A manual check with a voltmeter showed the -15v supply was within the required +/-10% tolerance. Other internal supplies were monitored as well by the computer and they never exhibited this issue. Below is a schematic of the components involved, what I found and the interesting solution.

What this schematic is showing is three resistor divider circuits that divide down the +15, +5v and the -15 volt supplies to feed the A/D converter inputs (there is some additional filtering and protection not shown). This A/D converter can resolve a range of 0v to +3v signal thus the need to divide down the voltages before conversion. Additionally the -15v rail voltage is "translated" to a positive voltage by raising it with the +5V. The code in the computer then reads the A/D converter and looks to see if the value read is out of +/-10% tolerance.

There is a problem though with the -15v divider circuit. Do you see it? The -15v circuit is dependent on the +5v voltage as well. So for instance if the +5v rail is 8% too high (within spec), and the -15v signal is 8% too low (also within spec), the resulting divided/translated voltage may be more than 10% out of whack (this is a highly technical term).  The unit would then erroneously fault. 

I found a simple software change that put this issue to rest. We need to first read the +5 rail voltage.  Then the -15v rail voltage is read and we subtract out the component of +5v rail voltage from the -15v reading.  To do this requires some math and Kirchoff's voltage law. 

V_AD3 = V_+5v - ( V_+5v - V_-15v) R₅ /( R₅ + R₆)

Next we have to solve for V_-15v :

V_-15v = (V_+5v - V_AD3)(R₅ + R₆)/R₅ + V_+5v

I implemented this equation in the firmware of the computer and it worked like a charm!  I implemented it, but did not consider this solution to be optimal. In subsequent designs I made sure we did not use the +5v supply to translate the -15v supply voltage, but for this product this software solution had several benefits:

• Easy fix for units already in field.  No recall.
• No waste for already manufactured inventory.  We can use existing stuffed pcb's.
• We did not need to delay an already delayed product.
• No additional cost for the pcb NRE.