The Four Costs of System Conversion

I wanted to write a short addenda to my last note on System Administration Principles.

In that first part I introduced:

Principle 1:  The Repair or Replace Decision

Principle 2: The Cost of Additional Computers

Principle 3: The Cloud Advantage

Principle 4: The Dual Platform, Dual Browser Safety Net

Principle 5: Legacy Code

Principle 6: Redundant Systems

Principle 7: Backups and Archives

Principle 8: Monetizing Content

Principle 9: High Value vs. Low Value Content

Principle 10: The Make vs. Buy Decision

With these principles in hand, and a good dose of common sense, I want to broach the important topic of System Conversion.

System Conversion: A change in method or process of accomplishing work or doing business.

Examples of this are changing: 

• operating systems - e.g. Windows vs. MacOS
• internet service providers - e.g. ATT vs. Cricket
• communications providers - Comcast vs DirectTV
• changing email programs - Outlook v.s gmail
• hardware platforms - Dell vs. Apple
• search platforms - Google vs. Bing

More specifically let's say you are unhappy with your email service provider - I myself  want to change service providers, but there is a time and monetary cost to doing so. When I use the word cost, I am referring to  time, space, and money (energy).

It is essential to calculate the cost for two possible scenarios:

1) The cost of continuing to do things as you currently do them.

2) The cost of doing the same thing a different way.

There is an additional cost:

3) The cost of doing the conversion itself.

To make matters worse, there is a fourth, hidden cost.

4) The cost of maintaining two systems while you transition from the pre-conversion system to the post-conversion system.

Doing something a different way does not mean doing it a new way - but it should always mean doing it a better way.

If you calculate these costs, before you perform a system conversion, you are more likely to make a good decision.

Consider the cost of an email system conversion. I am unhappy with Outlook.com because it takes a full thirty minutes to receive a new email message. When I am "cooking" I operate on a considerably shorter time frame than 30 minutes. Even 30 seconds when I am thinking about something can seem like an unproductive eternity.

I used to run Outlook 20XX on Windows YY on my PC ZZ. Then I migrated to Outlook.com as support for Windows was beginning to wane. Now I am almost completely transitioned to Outlook.com and my email is horribly slow. To examine what my options are I visited emailclientmarketshare.com. At this writing the marketshare is this:

I have an archive of 52,000+ emails reaching a back to 1996. So storage space is an issue, as are stability, longevity and ease of converting yet again.

My wife taught me a shopping trick. When she sees a set of similar products in an aisle and one of the products is nearly sold out, she buys THAT product. It is sort of a reddit, "wisdom-of-the-hive" approach. But that trick won't work here because there are more considerations than just the client.

After counting my costs the lowest cost is for me to continue what I am doing and letting Outlook 20XX coexist with Outlook.com. But I am simultaneously transitioning to the MacBook Pro as a backup platform which gives me portability. After as season it may become much easier to perform the email System Conversion.

Mixed Signals

Radio, digital or analog, is all about mixed signals. There is a popular saying, "I'm getting mixed signals", which implies confusion. But in radio, mixing is everything. Without it we cannot send or receive audio signals over radio frequency connections that use space itself as the connecting "wire".

Audio signals can be electronic, meaning they are represented by electrical signals flowing in wires. They can also be mechanical as in solids, liquid or air when we hear a sound or make a sound by tapping the table. My wife is tapping the table right now because she wants to go somewhere, but I'm hoping to finish this before she leaves me. Anyway:

Radio signals can also be represented by electrical signals flowing in a wire, or by photons of long wave light shining through the air and transiting the atmosphere, the ionosphere and ultimately empty space.

To make any kind of a useful radio we have to combine an audio signal - or a digital signal, as in where are my internet packets - anyway I was saying, we have to combine an audio frequency signal with a radio frequency signal prior to sending it out on the airwaves. Exactly the opposite occurs in the receiver:

goo.gl/09wN6m

To conclude this preamble, make the following mental note: When we mix one audio signal (the drums) with another audio signal (the guitar) on a mixing console in a recording studio, we are ADDING the signals together.

When we mix an audio signal with a radio signal in a radio prior to transmitting them, we are MULTIPLYING the signals together. This is called MODULATION. It comes in various flavors, AM, FM, PSK, but in all of these, the signals get multiplied to get on the air. When the signals encounter a receiver they are DEMODULATED, also by MULTIPLYING them with a radio frequency signal.

Mixers in Recording Studios Add Their Inputs

Now here's the bizarre and very cool thing that makes radio possible. When you mix two audio signals by adding them, the resulting signal is just the sum of the original signals, drums and guitar or what have you. Run with it. But when you mix two radio frequency signals by multiplying them the result is TWO signals. 

Here comes the good part. The product of two signals is two new signals - one at the sum of the original frequencies and one at the difference. So its like you can't get away from the sum thing. Multiplying them produces a sum (and  a difference which is just adding the negative). These sum and difference frequencies are called intermediate frequencies - IF for short - and they can be filtered and demodulated to discover the original signal be it audio, be it drums or guitar, or even my missing internet packets.

Since I am very interested in radio I was curious when mixing two signals what the issues were with regard to amplitude, frequency and phase.

Primarily I was concerned with the following. When we are receiving very faint radio signals from distant places like quasars or galaxies we have a very faint signal that is being mixed with a very powerful frequency produced by a local oscillator that enables us to move the sum or difference frequency to an intermediate frequency where we can grab it for demodulation (AM, FM, PSK or other). The whole mixing and unmixing thing is separate from the demodulation thing. We just have to agree before modulating what the demodulation scheme will be, so we don't even have to talk about that for this concern.

Long story short, I threw a couple of signals into my favorite geometric analysis program - Geometry Expressions™ - GX for short - written by the gang at Saltire Software - an AMAZING shop. Anyway here is what I found. Let's say the red signal is the local oscillator that you are tuning the signal in with. It can be as powerful as you want since you have control of it. Let's say the green signal (for little green men) is the signal that you are trying to pick out of the cosmic dust, litter, flotsam and jetsam. The blue signal is the mix of the two, and for this example we can see that it has about the same amplitude  as the little green men signal. 

From noise considerations, we don't want our red local oscillator to be any "louder" than the green signal, because we don't want any noise that may be in the oscillator to swamp out our fragile tiny signal in green.

From this we posit that the magnitude or strength or amplitude (however you want to say it) of the mixed signals is on the order of the strength of the LESSER of the two signal magnitudes. So if you want to pull in galaxy X, you better amplify it before you mix it, because its signal is only going to emerge from the mix as strong as it went in. The trouble with amplifying it before you mix it is that you amplify the noise too and one would like to avoid that. So you have to filter the INCOMING signal to only be in a band where you think useful information will be. This can be done starting with the antenna, if you know what frequency your information is going to be found on. This works fine for 89.1 FM because you know where the information will be. Bad for Galaxy X and the little green men.

One other thing if both signals are the same frequency (idealized case) and they differ by nothing but the phase of the signal (where in the clock cycle we started), then we get a DC offset in the resulting signal. This is usually blocked by a filter capacitor. It is an issue that must be addressed. Here is how it looks in GX:

Note that the signal strength in this case is half the strength of either signal. Making the case that when looking for little green men, its best to be as quiet as possible...

Ten Computer System Administration Principles

The Device Boom

With the explosion of tablets, smart phones, apps, fitness monitors, remote controls and personal computers, we have been thrust into being computer system administrators.

Breathe

Making matters worse the tools we use to manage these devices has not kept pace with their population explosion. Some vendors, who will go unnamed, have made matters worse by stripping out features for Home users that ease routine administrative tasks like backing up irreplaceable information, updating operating systems and migrating files. You know - email, photographs, documents, financial records, blog entries, homework and so forth.

A careful reading of these ten administrative principles can lead to new places. They can be combined to make new derived principles. Some are similar, but just different enough to warrant their own mention For others you can substitute the word "house", "car", "pet" or even "relationship" to derive additional benefit.

Principle 1:  The Repair or Replace Decision

After concerted good faith effort to fix a broken "computer", if the "computer" does not respond, then it should be replaced without mercy.

Principle 2: The Cost of Additional Computers

For each additional computer system that you maintain, your time is diluted by the total number of computers. So if you have two computers that you often use, you must spend twice as much time maintaining them. Three computers will require three times as much time to maintain. If you have five airplanes you can spend all your time and money maintaining them and none of your time getting to fly them.

Since time is money, this also pertains to computers you pay someone else to maintain.

This principle gives us one obvious thing and one surprising thing.

The obvious is that we want the fewest possible machines possible, that get the job done.

The surprising thing is that if we already have 10 computers, adding an additional one doesn't cost very much. This leads to server farms and the economies of scale.

But I am assuming you don't want to run your own server farm. This leads us to ...

Principle 3: The Cloud Advantage

It is more economical to use a computer in the Cloud, provided it is secure, than it is to maintain your own server. Every second your own server runs, the more of your time and money it saps as it becomes obsolete. I use Dropbox (DB) for this purpose. I can access everything I do from ANY computer ANYWHERE I am located. The first 5 GB is free and they charge after that. DB runs the server farm. DB does the backups. DB fixes the broken equipment. DB is the transparent utility.

Principle 4: The Dual Platform, Dual Browser Safety Net

After losing support on various platforms through the years I have learned:

Any software you buy or use should run efficiently on more than one platform, currently chosen from the set {PC, Mac, Linux}. Any web-borne software you buy or use should run in more than one browser, currently chosen from the set {Chrome, Firefox, Safari, IE}. The food chain alphas flip every decade or so, no matter who was on top to start with.

Principle 5: Legacy Code: If you have legacy code you have to keep the legacy computer running in both hardware and software. The same thing applies to code that is not dual platform or dual browser.

Principle 6: Redundant Systems: When running an important system, it is valuable to have an IDENTICAL hot backup waiting in the wings in case the primary system fails. Note that this principle balances the cost of additional computer principles like porridge and the three bears. By which I mean, 1 critical system is too few, 3 is too many, and 2 is usually just right. In fault-tolerant systems that have "votes", you want odd numbers of machines so you can take the best 2 of 3, the best 3 of 5 and so forth. This gets into RAID disk arrays and error-correcting memory schemes, so that fault tolerance can live at the subsystem level, at the box level, or both.

Principle 7: Backups and Archives:

Always have at least one and preferably two copies of any document in addition to the original. The copies should be on a different physical drive, and preferably in a different physical location. This can get unwieldy, see principles 1, 2 and 3 for guidance.

If you backup a file that has a virus, the virus will be in the backup.

Principle 8: Monetizing Content:

Any content, software, or data, that you are "spinning" on an available drive should produce value for someone. If it is never accessed or used, there is no use paying to spin it. It should move to archival storage which is a HUGE but separate issue.

Principle 9: High Value vs. Low Value Content:

If you have authored specific content, that content is of high value. If you did not author that content and someone else is spinning it, then you don't have to. Its value is lower because it is reachable elsewhere. If there is high demand for the content then you need a redundant system or mirror.

Principle 10: Make vs. Buy Decision:

Don't make something when you can buy that thing for less money and time than it would cost you to make it. You should also consider the cost of the environmental impact of the life cycle of the thing when tally your cost.

(Disclaimer: this content is offered without warranty of merchantability or fitness for any purpose, expressed or implied, which really should have been its own principle!)

Anding on the Backend

We use Boolean operations everyday without thinking about it.

Whenever we read, write, say or decide using the terms "and", "or", or "not" we have used a Boolean operation.Whenever we think about selection, inclusion, exclusion or negation, we have used a Boolean operation.When programming computers (stay with me for second) we often write phrases like:

if (conditionA is true) AND (conditionB is true) then do something.

When making routine decisions we might think something like:

IF (sky is blue) AND (temperature is warm) THEN go outside.

We can talk about what the terms sky and blue and temperature and warm mean later.

Not only can we write symbolic expressions, or think symbolic thoughts, we can draw symbolic pictures.

Where I am going is this. 

When we write or think in the way discussed above, we are doing our Boolean operations on the front end.

that is, "IF such and such (Boolean operation) THEN do something"

But we can also perform Boolean operations in our actions on the backend.

That is to say, "IF such and such then do something (Boolean operation)".

For example

IF (sky is dark) THEN (take umbrella) AND (go outside)

We can even have Boolean operations on the front and backend

IF (sky is dark) AND (temperature is cold) THEN (take umbrella) AND (wear coat) AND (go outside).

But forget all this for a moment. I said all that just to say the following. If we are doing an OR instead of an AND we run into a problem.

Whenever we ANDed two or more actions, it is understood that we did BOTH or ALL of them, whether sequentially or concurrently it doesn't matter right now. But when we OR two or more actions we are left with a curious nondeterminism:

IF (sky is dark) OR (temperature is cold) THEN (take umbrella) OR (wear coat) AND (go outside).

We are left with an unspecified decision. These decisions come up all the time in real life, and we often dismiss them, but they are an enormous source of decision noise in our lives.

I would like to dig in to both the mechanistic/automatonic view of this nondeterminism, and also the practical and real world view of it.

That's the deal!

Ode to an Orchestra Obscured

The concert is a time to stop and think, third row center C. My muse, my mizzen mast, cracks the rust off from above.

The conductor is drinking stratospheric air; a shining breath of freshness from Olympus and the rest.

I would write this ode for him alone - a focal point for sure, but my problem runs much deeper.  I am in love with an entire orchestra, whose story I yearn to know. Each player a dedicated life of practice and disposition, each a bag of presents, tricks and tips.

The conductor is of agile build, a shock of hair, and sweeping gestures demanding unseen witnesses to rise up and pay attention. A genius, I will give him that, not worn upon the sleeve or spoken to a mirror. It is in the light of piercing eyes, one as called then chosen. Enough of that. I’m glad for a wooden bar to brace a treacherous fall from grace. His wooden bar, it blocks my view, accepted for its safety, lest rising  up upon his toes he steps back into the void.

Last night’s concert - I could write about them all. But one must choose a single instrument. I choose wind and brass and key and string!

As for strings, the blackened box was full, sitting grandly in the foreground, performing surgery on my open heart. A keyboard surgeon, a pair of hands, an arc in smooth curved motion with ebb and flow, easing up and down. There, a note, a stroboscopic sleight of hand. Each one, an evening out, a beginning, middle and an end.

Yet even in perfection there was a problem, that not pianist,  nor conductor, nor orchestra could stand. The problem was the piano that I looked upon - like one would look up to a mountain knowing there was nothing I could do.

It was in freefall overhead, dropped from a skyscraper after being lugged up endless stairs. Descending like an anvil, whistling in freefall, as I walked unknowingly on the sidewalk of ailse C, carrying flowers to honor one musician, then another, and then finally them all. Not knowing that in a fleeting moment, I would hear that sound.

Gazing upward I got to know the belly of that piano like a prisoner doing time. Marking off the days, I could see the milking of its pedals, not the twinkling of its strings, not its long wrapped copper notes, nor its trebles strung in threes. I could not consider the infamy of its even-tempered tuning, the low notes flat, the high notes sharp - let the fretless understand.

By this massive ruse in ivory my orchestral view was blocked, choked from light, a plant on the jungle floor doomed by rules I did not make to die in desolation. You ask, "What was it pray tell you wanted to see?" Deep breath. It was the bridge, the bow, whether near or far, whether dinner or dessert, a salad or a feast. The scoot, the slide, the brawn, the bangs, the beard, the chin, the pad. The chrome, the brass, the wood, the grain, the warp, the weft the weave. The letters, the luthier, the make, the model. The hope, the skill, the miss!

But I could see! A constellation of creatures, each with six legs! The back legs sets of four, then slats of wood then two thick ones dark with wider feet, some fronts were striped forms. Four were always still and thin, two were always moving. Nourishment perhaps?

I vote for fewer legs, but alas I have no vote. I can only offer a petition. Strike that. I am a creator of the most demanding kind! Foisting tyranny on the innocent on long-furled scrolls, untied at a moment's notice! No prisoners of war, no hostages begging for mercy, just an ode to an orchestra obscured.

What I Learned from Bill the Buddhist

Whatever the endeavor:

"If you practice, you will improve"
"People get bored"
"People are impulsive"
"People think half thoughts"

A guy is going 90 MPH in his car.
He opens the door and says, "I'm going to jump out".
He thinks a half thought.
He doesn't think about what will happen when he jumps.
-----
I met Bill the Buddhist while walking around the Forest Heights track. I was melancholy and decided to start a conversation to distract myself.

I pointed out that the track, open to the public all year round,
was surrounded by barbed wire - an interesting contradiction.

Then I observed, "Isn't it interesting how people who don't know each other never walk or drive side by side. One will speed up, the other slows down. They space themselves apart as far as possible."

Bill the Buddhist then told me a blue-haired woman pulled up  beside a friend of his in her car. "Like Marge Simpson" I said. "Yes", he replied going on, "She looked at his friend in his car, looked at her door and promptly, assertively, locked her doors. His friend then looked down at his door and pushed HIS lock down, locking his own doors." We agreed this was funny.

Bill had more stories such as "Feral children". "If you don't get to them by the time they are seven, they never socialize." I thought rescuing feral children was an interesting and under discussed topic.

He talked about "same substance", that people are all the same, male, female, old, young, etc. "All flesh" he said. I said, "Yes, but if the flesh is the body, they have different drivers."

He talked about social norms and people and how people are affected by the norms of other people. When people are isolated they become feral again and will do strange things.

He talked about how he went to live in Japan for a year and a half. He said, "You don't really learn a culture in that time,
but it opens your eyes on your own culture." I asked him, "How so?". He said something profound but I forgot it.

Then he talked about how they drive on the "wrong" side of the street in Japan. "Like in Britain" I asked? "Yes" he replied.
He thought it profound that they chose left and we chose right.
I did not. I thought it was arbitrary and insignificant and said so.

Then we started talking about bifurcations.
Bifurcations are where a choice has to be made and 
one culture chooses one way, and another chooses the other way.
Like which side of the road to drive on, which affects
on which side the steering wheel is. Bifurcations are everywhere.
Your liver is on the right, but it could have just as well been on the left, and so on.

We talked about language bifurcations.
How Egypt had a language based on pictures and we had one based on the sound of words. Picture languages are great for nouns and bad for verbs. We talked about the Babylonians, the Rosetta Stone, the Indo-Europeans and things that could have gone either way and did, or did not.

Then we started talking about communication. I told him, "Communication is symmetric if A can talk to B and B to A."
Before that we talked about Godel and undecidability and how Godel starved himself to death when his wife left and couldn't fix him dinner.

Before that he said the inner part of a person is infinite.
I told him, "I don't feel infinite on the inside, I keep bumping up against the stops."

Then I mentioned that communication is reflexive if you can talk to yourself. We mixed this in with perception, which is a kind of self-talk. He lamented the incompleteness of communication and mentioned Iran as an example. I said communication is very complete if you do it properly.

Then we made up a game. A bifurcation game.
Each of us writes down one of two symbols, and gives it to the other person. The other person then writes down their symbol and gives it to the other...
The person is not obligated to read the symbol they received before they transmit their symbol. This special case is called "Not listening".

Then Bill the Buddhist said these are his principles:

Whatever the endeavor:
"If you practice, you will improve"
"People get bored"
"People are impulsive"
"People think half thoughts"
A guy is going 90 MPH in his car.
He opens the door and says, "I'm going to jump out".
He thinks a half thought.
He doesn't think about what will happen when he jumps.


-----
I met Bill the Buddhist while walking around the Forest Heights track. I was melancholy and decided to start a conversation to distract myself...

Circuits As Proofs: Episode 1

Electronic Circuits As Mathematical Proofs
Episode 1: Resistance is Futile

L. Van Warren MS CS, AE

Warren Design Vision
 wdv.com

Friday, February 3, 2012

Introduction

I want you to forget for a moment that you are reading this as a consequence of digital technology.  I want you to forget that the screen it was written on was imaged as ones and zeros running through a multitude of logic gates at high speed. I want you to forget the mountain of computational complexity we use everyday as our lofty perch.

A Lofty Perch

The Naked Truth

At various points in school and life you may have encountered a mathematical proof. For many they are tedious, boring and purposeless arguments that seem to serve no purpose but their own. I hope to convince you that without them, the digital revolution you now enjoy would not have happened. Further, I hope to give you a way of seeing proofs as never before, as everyday objects,  familiar as bread and butter that you can consume to increase your understanding.

An Everyday Yet Monumental Example

When one enters a dark room, a common behavior is to flip on a light. A switch is thrown, and usually, but not always, the light turns on. Most of the time we proceed without giving this monumental event a second thought. But I tell you, that when and if that light came on, at that very moment, you proved something. You proved something mathematical. You proved something real. You proved that the light, and everything connected to it," worked". This is profound.

Not finished with your monumental act, you did something just as stupendous. When you left the room, the switch was thrown again. Usually, but not always, the light turned off.

If you flipped the switch AND the light came on, you proved that:

1)    the switch "worked", it served as a human control between an OFF state and an ON state.

2)    the wires from the switch to the light were connected and conducted electricity.

3)    the light itself "worked", that its filament, tube, or junction converted electricity into light & heat.

4)    that the power company, however complex, was intact and functioning.

If you flipped the switch and the light didn't turn on, you proved that one or more of the above statements was false, an equally majestic and complex treat of logical possibility.

Possible Explanations

It is possible, if you flipped the light switch, that something else happened.

It could be, that at the moment your hand touched the switch, you were abducted by magic aliens who connected your mind to an illusion machine that made you believe that the light turned on or off.

A Futile Resistance

This explanation violates common sense. It seems highly improbable. For Kurtnestetic incompleteness, we will consider such explanations unlikely at our current scale and existence. The magic aliens will be left in the care of our more mechanical readers, quantumly speaking

Situation as Circuit

Using this free download we sketch a circuit that represents our situation:

We have already committed several acts of abstraction, having:

·       replaced real world objects by symbols in a graph

·       chosen a voltage source  to represent the power company

·       chosen an LED to represent the light "bulb"

·       sized the components so they didn't blow up

The resistor is along for the ride. It represents inefficiency and injustice in the system,  such as "wires aren't perfect conductors" and poverty. We could have a separate resistor for each of the three original wires, but each time we add a resistor we must add a wire. This tendency of things to become more complex before they get simpler is called, "intermediate expression swell". Just how bloated things get is an interesting problem we will sidestep for now.  Pushing all three resistors through the circuit and placing them side by side allows us to replace them all by a single resistor. All the losses are then treated as single equivalent loss, adding another act of abstraction to our efforts!

In doing so we invoked the rule:

"Resistors in Series Add Linearly"

R_equivalent = R₁ + R₂ … + R_N

Having transformed our little world into an "equivalent" circuit, we must define what "working" means. "Working" means Proven, Demonstrated, QED.  We define "working" to mean that current is flowing around the loop in sufficient quantity and with sufficient potential to turn on the light. This is something we can measure in both worlds, simulated and real. We can measure voltage, we can measure current and we can measure whether the light is on or off. The single measurement of whether the light is on, abstracts away the need for two measurements, voltage and current. Aside: I always measure the voltage of household batteries before and after installation. It is incredibly informative. Measurement of current requires a meter alias ammeter to be placed in series with other circuit elements. Current flow is measured in units of Coulombs per second, or Amperes, much as water flow is measured in liters per second. It is represented by an arrow, labeled here "Meter de Courant" in a twisted French homage to the German American mathematician Richard Courant.

When the Hardware meets the Footwear

Two pictures conclude the example. One picture with the switch OFF, one with the switch ON. QED

OFF:

ON:

The astute reader will notice that the current is actually flowing more voluminously in one direction than the other. This happens too fast for us to see, so we think that the light is on, when it is actually pulsating. Because the light is an LED, it only shines when conducting current in the forward arrow direction. The back-flow happens because real diodes are naughty. They leak. Current flow is taken as positive when traveling from positive pole to negative pole of the source. The squiggly "current-limiting" resistor is sized to keep the LED from burning up like a comet and destroying our pretty little world in one puff of smoke, also known as the smoke test of a proof. The tiny fence on the arrow of the diode implies that current is not supposed to flow backwards even though it does. The act of proof, the light turning on when the switch is thrown, isn't affected by these details. They do however suggest deeper and more interesting issues are afoot.

Acknowledgements

This work is dedicated to parents, professors and friends that inspired it.
Illustrations are linked to their respective sources.

© 2012 · Warren Design Vision

An Unexpected Circuit

Introduction

I am going to write a short paper. It's going to be a fun paper because like you, I don't know how it is going to turn out. That should make it interesting for both of us. Today's topic is a little weird -  it arose in a rather offhanded way. I wanted to write about it because it was like stumbling on buried treasure. I am going to write in the first person because it is easier for me to express myself that way.

The stupefying beauty of this is that if I don't succeed, it is incredibly possible that you may. If I happen to succeed, we both get a useful tool.

Background

The problem is this. I need 50 Volts of Direct Current in a lightweight and portable package. 50 VDC, not 6, 9, 12 or 24. It is okay if I have to use an outlet or extension cord but I need 50 VDC. One more thing - I need 600 Watts of it. Because power is voltage times current there will be 12 Amperes flowing through a wire somewhere. Oh and one last thing -  I need this 50VDC with 600 Watts of power constantly available on the shortest of notice, say 1/20,000^th of a second and also in a sustained manner.

Please don't hang up when you read the following statement. Stay with me, it will be worth your while.

An electronic circuit is a mathematical proof.

A circuit is a proof. It is a proof in an ideal sense with idealized components. It is a proof in a real sense with real-world components. If a circuit works, then the system that it represents is true. Many mathematical and mechanical systems have equivalent circuits. We will revisit this powerful idea.

Possible Ingredients

Lead-Acid

50 VDC is a weird number to want. I could use four 12 Volt lead-acid batteries, settling out of court for 48 Volts, knowing full well Lead-acid batteries are not 12 Volts they are 12.63 Volts. I would settle for 50.52 VDC. Lead-acid batteries have two problems though, lead and acid. The first problem, Lead, is toxic, and more seriously, it is heavy. I don't like heavy things. I don't like lugging them around. The second problem, Acid, is also toxic, and more seriously, it burns stuff. I don't like burns and that yucky corrosion that forms on car batteries. So let's just drop lead-acid.

Lithium

I could use thirteen 3.7 Volt lithium-ion polymer batteries, settling for 48.1 Volts. Lithium batteries have two problems, cost and explosion. I don't like things that are too expensive for widespread distribution. Also they explode if they aren't charged carefully. I can handle a small battery exploding. That might even be fun. A battery with nearly a horsepower is not small and neither is its explosion. So strike lithium.

Step Down Transformer

I could use a 120 VAC step-down transformer to transform household current to the lower AC value and then rectify and filter that into 50 VDC.

Two problems emerge. Even ordinary 120 VAC step-down transformers are expensive and they are heavy. I already mentioned my dismay with lugging expensive stuff around. The astute reader will note that this is a lie. I mentioned my dismay with lugging heavy things around and my dismay with things that cost too much, but I did not mention my dismay with lugging expensive stuff around. In that subtle distinction, there is meaning. You might think I don't like lugging expensive stuff around because of theft, and that would be true, but that was not the point. The point was, I never said that, but if one wasn't careful, one might assume I said it, and leave the encounter thinking something was said that wasn't. The structure of this argument is similar to a circuit we will discuss in a moment.

Now since I have ruled out batteries or transformers I need what is called a switching power supply. Switching power supplies are cheap and abundant, but they have a problem that intimately affects their use. They create NOISE. Electrical noise. Electrical noise bothers me more than weight or cost.

MOSFET Modulation: Smooth, Quick, Efficient

What I would like to do is somehow sample 120 VAC in a smooth, quick, and efficient way. Then I can rectify it into 50 VDC and filter it according to standard practice. So now, we have an apparently simpler problem than 600 Watts of 50 VDC. I just need 600 Watts of 50 VAC. Ignore that because of efficiency more than 600 Watts is needed. Ignore that because of voltage regulation, a more than 50 VAC is needed. Settle for 600 Watts of 50 VAC. If it is made from 120 VAC smoothly, quickly and efficiently the rest is cake.

What if instead of brutal switching, which creates far-reaching wideband racket, we gently turn something on and turn something off in a smooth and efficient way?

This circuit does that… partially. The control gate of an N-Channel MOSFET is fed with a version
of the same signal we are trying to clamp to 50 VAC. I say, "clamp" because when the voltage goes over 50 VAC we don't want any part of it. We especially don't want to accept this higher voltage and in turn dissipate the excess energy as noise or heat.

We can exactly tailor how much of the positive going side of the 120 VAC makes it to the voltage probe by adjusting the values of R1 and R2. If they are equal, we get 50 VAC on the positive going side of the wave:

So just like that we are halfway done. The signal is about 50 VAC on the top. In the figure above, both the 120 VAC signal and the output signal are drawn.

We would like to do the same thing on the bottom, so let's start simple and build a circuit that just does the opposite of the first one. Maybe then, we can combine them to get what we want. A circuit might do the "opposite" is one that uses a P-Channel MOSFET, the "opposite" of an N-Channel MOSFET:

After trying some different configurations we run it and get what we want –the negative side of the 50 VAC is present:

Now perhaps we can combine the two that will clamp both sides of the waveform to 50 VAC.

Running the simulation, we obtain:

We have achieved our objective. A real circuit might have some additional parts for protection, but this shows that the configuration is valid.

What I did not show you was the search process that a reasonable person might have employed to come up with this configuration.

If you look closely at the schematics, you will notice that MOSFETS are not commutative.
With the gate in the middle, they can be put in forwards or backwards.

This gives rise to four possible circuits, since there are two MOSFETS in use, and only one circuit topology will produce ("prove") the correct waveform.

In addition, the MOSFETS need not live on the positive supply, they could also be placed on the bottom of the load, but because of symmetry, this would still prove the correct waveform:

So now, we have eight possible circuits of which two are equivalent and correct and six are incorrect.

Additionally we can swing one MOSFET around to be on top of the load and one on the bottom:

Which produces the following output:

Swinging the P MOSFET around instead of the N MOSFET produces a similar incorrect outcome. That makes 2 of 10 possible configurations correct.

We can increase that by 4 more configurations by pivoting each MOSFET in the swung around configurations, which also produce incorrect outcomes.

If I have counted correctly, 2 out of 14 configurations are correct. But now we are not doing circuit design. We are searching over a space of possible connection topologies for a topology that proves (produces) the waveform we sought. Each of the topologies proves something, but only two proved what we set out to prove.

Conclusion

We have succeeded! It was discovered that we can produce (prove) the desired waveform and discovered two topologies that were satisfactory. The circuits we have produced are equivalent to other systems that have similar algebraic or mechanical characteristics. These equivalent circuits and their topologies not only model the equivalent algebraic and mechanical forms, they model the search process we might employ to discover useful circuits along with its combinatorial growth.

Easter Egg

I mentioned above that "protective circuitry" was needed to take this synthetic design into a practical unit. It turns out that the MOSFET gates need their voltage clamped to around +/- 10 volts. So the  working design looks like this. Twelve parts drawn from 5 types, not counting power cord, fuse and load. An ounce of material you can hide in your hand. Two MOSFETS and power diodes in the top section, all in T-220 cases on a heat sink to handle 12 amps. The “Logic” section on bottom, runs on 6 milliamps that turns on the MOSFETS in 7 microseconds. 576 Watts. $9.15 worth of parts, the front-end of a $600` power supply. Those savvy in the trade will notice that this is a smoother version of an H-Bridge circuit used to control motors, so they don't jerk as much when doing, "the robot".

The file with the component values is available for $9.95 if you send me an email at the address on the left. It uses TINA SPICE from TI.

Acknowledgements

Brian Beckman, private communication.

References

As hyperlinks throughout the text.