Remember that piece I wrote about the “Universal Secondary Winding” ?

This is “Part 2″ thereof, meaning … The LAST high voltage toroid that you shall EVER purchase.

Interested?

Read on.

About the Balanced Ternary Numbering System (Say .. about WHAT ???)

But let us go slow on this. As a refresher: OK, so we have a certain number of “secondary windings”, and obviously, each winding can be connected with some other winding in series.

**http://hiend-audio.com/2013/12/14/the-universal-secondary-winding**

BUT ……

But the “series” connection can take place in any of the two following ways:

a). In-Phase, such as that the “end” of the first winding can be connected in series with the “beginning” of the second one, so that you would then use the “Beginning_of_no_1″ and the “End_of_no_2″ winding as the final tappings of the resulting winding,

OR

a). Anti-Phase, such as that the “end” of the first winding can be connected in series with the “END” of the second one, so that you would then use the “Beginning_of_no_1″ and the “BEGINING_of_no_2″ winding as the final tappings of the resulting winding.

Please note the subtle, but decisive difference between the two scenarios.

In the first scenario, you literally “ADD” the two AC voltages together, sort of by simply “extending” the number of turns within the first winding by the additional number of turns of the second winding. {{ This is like taking an “exit” too early off a motorway … What do you do ? You simply return back to the motorway and drive further on to the next “exit” – the one of interest }}.

But in the second scenario, you essentially do the OPPOSITE: you “Substract” the number of turns of the second winding from the number of turns of the first winding.

Essentially, using this method, you can either ADD or SUBSTRACT the AC voltages of the two independent windings.

Free Lunch ?

Not quite. There is a small penalty to pay in the latter (i.e. subtraction) scenario.

It is the penalty of a slightly increased internal DC resistance (yes, “Resistance”, not “impedance”) of the resulting winding. It is as though the current must first flow through a certain number of “superfluous” turns ( the ones that are “too many” ), … but then back tracks along the same tracks, has to do a “U-Turn” and then flows “backward” through the second winding, in antiphase, so as if it “nulls” off a number of excessive turns of the first winding. But doing so, it STILL flows through an extra, additional length of WIRE.

{{ This is like missing your proper “exit” on a motorway. What do you do? You drive along forward, find a place where you can perform a “U-Turn”, then you turn back and backtrack along the same route, until you get back to the exit of interest. But having said that – the extra mileage and the burnt fuel is the penalty for such a solution }}.

So, your current, as it does it’s U-Turn (going from the “End_of_Winding_no_1″ back to the “End_of_Winding_no_2″) actually also incurs a penalty in the form of the extra resistance of the “extra mileage” of wire that it needs to travel. Wire, that has additional resistance. Having said that, the penalty, at least in the case of tube applications, is not critical. These few additional ohms will be more than adequately compensated for within your capacitor bank of the filter after rectification.

So, in essence, we now have a method of both adding up, as well as subtracting voltages.

Going back to that magic sequence of windings, from the first part of this blog entry, we have the following:

1 ( = 3 to the power of zero)

3 ( = 3 to the power of one)

9 ( = 3 to the power of two)

27 ( = 3 to the power of three)

81 ( = 3 to the power of four)

243 ( = 3 to the power of five)

….

What you see here is in essence a trinary coded code of the respective numbering system.

This is NOT a “Base_Ten” numbering system (…per analogy to the number of fingers on our hands that we have to count with).

This is also NOT a “Base_Two” numbering system (… per analogy to the number of “fingers” that our computers use to count with).

This is a “Base_THREE” numbering system.

Albeit a very “special case” base three numbering system.

It is a numbering system that uses three possible digits, but instead of using the “typical” ones, i.e.: {0, 1, 2}, it rather uses “funny” digits, and mainly: { -1, 0 +1 }.

This is called a *** BALANCED TERNARY *** number system.

*{{{*

*if you do not know what I am talking about, read the following additional literature:*

*http://en.wikipedia.org/wiki/Numeral_system*

*http://en.wikipedia.org/wiki/Positional_notation*

************* http://en.wikipedia.org/wiki/Balanced_ternary ***************

AND/OR: Imagine that instead of counting on **10 fingers**, you have only **THREE fingers** at your disposal, … such as an Extraterrestrial, an ET, … or Some…THING :)

*}}}*

Why three as the basis of our toroidal transformer numbering system ?

Simple: it just turns out that way.

Look at it this way: To obtain the number “1″ – you simply use the digit 1. The next logical digit position must be a “three”, because by using it, you can obtain “2″ by simply subtracting a “1″ from a “3″. So there you have it. The rest is pure consequence.

You get the “4″ by adding “3″ + “1″.

You get the “5″ by … using the next numbering system position, i.e. the 3 to the power of 2, meaning a “9″.

You get the “5″ by … subtracting a “3″ and subtracting a “1″ from a “9″.

So there you have it.

THIS IS THE LAST TOROIDAL TRANSFORMER, that you shall EVER have to purchase for any and all of your DIY projects:

It’s secondary windings shall have the following turn counts:

3^(0) = 1

3^(1) = 3

3^(2) = 3 * 3 = 9

3^(3) = 3 * 3 * 3 = 27

3^(4) = 3 * 3 * 3 * 3 = 81

3^(5) = 3 * 3 * 3 * 3 * 3 = 243

3^(6) = 3 * 3 * 3 * 3 * 3 = 729

3^(7) = 3 * 3 * 3 * 3 * 3 * 3 = 2187

3^(8) = 3 * 3 * 3 * 3 * 3 * 3 * 3 = 6561

….. Ooops ! before we get TOO optimistic on this, please do not forget that we are now talking about a possible voltage of close to **984,1 Volts AC !!!!! **

How come ? Well, remember what I said at the beginning, that each turn is approximately 0,1 Volts AC, in my part of the jungle, at least.

So, here you have it: With nine digit positions (ranging from 0 to 8) to count with, within your base_THREE numbering system, the maximum number that you can obtain is “111111111″, which in a BASE_THREE system is equal to the following number of turns:

1* 3^(8) + 1* 3^(7) + 1* 3^(6) + 1* 3^(5) + 1* 3^(4) + 1* 3^(3) + 1* 3^(2) + 1* 3^(1) + 1* 3^(0) =

= 9841 turns.

Multiply that by 0,1 Volts / turn, and you get the 984,1 Volts I told you about earlier.

**BEWARE:**** ****The INSULATION between each of these INDIVIDUAL windings needs to be capable of withstanding at least 3 TIMES the maximum possible voltage that is possible to obtain by any and all combinations of these windings. ****Each insulation must hence resist at least 3000 Volts AC. **

I suppose that it would be safer for you if you skip a winding or two from this crazy scenario.

**If you skip just ONE winding of the lot, skip the 3^(8) for example, and throw in the towel at 3^(7), you shall still obtain a maximum voltage of around 3280 / 10 = 328 Volts AC, which should cover something like 99,9999 % of any and all of your present and future anode voltage needs, as after rectification this would provide you with something like 460 VDC. **

The insulation between any and all of these windings, as well as between them an the PRIMARY winding, is somewhat relaxed, it suffice that you use a 1500 VAC insulation between them, which shall provide you a reasonable amount of overkill protection (1500 VAC vs. 328 VAC max. attainable).

If this 460 VDC seems a tad low for any and all of your future purposes – keep in mind that the 0,1V per turn is a slight simplification. In reality it may be just a bit higher, say 0,12 or even 0,14, depending on the make of toroidal transformer, the type of it’s core, etc.

The “precision”, or raster, or “voltage step INCREMENT”, within this “winding-adding” and “winding-subtracting” game, is represented by the least significant number system position used. Specifically, this could be a resolution of 3^(0) =1.

Meaning ONE.

One single TURN.

One single turn, which roughly represents 0,1 VAC. That is 100 milivolts of AC, constituting a single step, on a scale ranging from 0,1 VAC up to 328 VAC. Very nice.

Still, if the 328 VAC border, as represented by 8 windings {the biggest one: 3^(7)}, is too little for you, you can TRADE IN ”resolution” in exchange for “extended range of voltages”.

You can simply “scale up” the whole system by multiplying each and every turn count by two, such as in this following example:

2x 3^(0) = 2

2x 3^(1) = 6

2x 3^(2) = 3 * 3 = 18

2x 3^(3) = 3 * 3 * 3 = 54

2x 3^(4) = 3 * 3 * 3 * 3 = 162

2x 3^(5) = 3 * 3 * 3 * 3 * 3 = 486

2x 3^(6) = 3 * 3 * 3 * 3 * 3 = 1458

2x 3^(7) = 3 * 3 * 3 * 3 * 3 * 3 = 4374

2x 3^(8) = 3 * 3 * 3 * 3 * 3 * 3 * 3 = 13122

This way, you shall achieve a voltage range with a maximum possible value of 656 VAC {{ in case of 8 windings, with the biggest one: 2x 3^(7) }}, or 1968,2 VAC {{ in case of 9 windings, with the biggest one: 2x 3^(8) }}.

The only drawback of such a “scaled up” scenario is that the “voltage resolution”, has now became 200 mV instead of 100 mV. Two times “coarser”, but still a very accurate scale. No problem. Peanuts.

OK. But at this point, one would say: ”8 windings ??? – this guy is crazy …. “.

The more so, one could say: ”9 windings ??? – this guy is even crazier still … “.

Well, if you do not like such an excessive number of windings, just simply get rid of the trivia: Get rid of the “smallest” windings, such as the ones represented by:

3^(0) = 1 turn,

3^(1) = 3 turns,

3^(2) = 9 turns,

… and hence remaining with these following, “reasonable” windings:

3^(3) = 3 * 3 * 3 = 27

3^(4) = 3 * 3 * 3 * 3 = 81

3^(5) = 3 * 3 * 3 * 3 * 3 = 243

3^(6) = 3 * 3 * 3 * 3 * 3 = 729

3^(7) = 3 * 3 * 3 * 3 * 3 * 3 = 2187

3^(8) = 3 * 3 * 3 * 3 * 3 * 3 * 3 = 6561

Now we see that we have limited the number of windings to only SIX. Please also observe that we have traded in on the “voltage resolution”. The smallest “step” or “increment” by which we may now navigate our voltage maps is the value of 2,7 Volts AC (=27 turns), which, in terms of an ’Anode Voltages’ scale – still seems to me as a very “fine” step change.

If, on the other hand, you are considering the “scaled up” version, meaning with the “2x” multiplier for the respective winding counts, the smallest “step” voltage shall become twice as large, meaning 5,4 VAC – which still represents a reasonable step value.

On the other hand, if this is too “coarse” a step – simply return back to the 2x 3^(2) = 18 winding, i.e. to the 1,8 VAC step. Simple. One additional “small” winding, providing you with a finer “resolution”.

Before you run off to start ordering your universal toroidal transformers, please also bear in mind that it is fairly easy for you do “add” your own windings, provided that the center hole is not filled up with some epoxy resin or other filling. It may simply make more sense to order the “higher” turn count windings, and then add to that your own DIY windings. Executing a winding with 27 turns is peanuts, to say the least …

Either way, whichever way you choose to implement this transformer, it shall, most probably, be the LAST high voltage transformer that you shall ever purchase for your various DIY experiments. Therefore, …. the DONATE button is on my front page, as a form of gratification from you to myself, in return for this wonderful idea that I told you about here.

Happy Holidays ! (as the Americans would say, or …)

Merry Christmas ! (as the Europeans would say).

Cheers, (as myself would say).

Ziggy.

P.S. …. Here is a nice extract from the Wikipedia:

“…. Balanced ternary uses a base of 3 but the digit set is {-1,0,1} instead of {0,1,2}. The “1″ has an equivalent value of −1. The negation of a number is easily formed by switching the ”-” on the 1s.

{{ zjj: this is us subtracting a winding instead of adding it !! }}

This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight.

Weights of 1, 3, 9, … 3^{n} known units can be used to determine any unknown weight up to 1 + 3 + … + 3^{n} units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with 1, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight *W* is balanced with 3 (3^{1}) on its pan and 1 and 27 (3^{0} and 3^{3}) on the other, then its weight in decimal is 25 or 1011 in balanced base-3. (1011_{3} = 1 × 3^{3} + 0 × 3^{2} − 1 × 3^{1} + 1 × 3^{0} = 25). ….”

(C) zjj_wwa, 2014.