# Getting 240 volts from 208 volts

Discussion in 'Electrical Engineering' started by [email protected], Nov 29, 2006.

1. ### Guest

The need to get 240 volts instead of using 208 volts for 240 volt
loads is not what I will be discussing here. If you want to discuss
the viability of using buck/boost transformers to get that 240 volts,
either for delta connected three phase loads, or for single phase

What my calculations came up with is a wiring arrangement using the
common 240/120 to 24/12 single phase buck boost transformer to get
nearly the correct voltage. Actually, it would involve 3 of them.
To understand what I am going to describe, you should already know
how a buck/boost transformer is used in an autotransformer setup to

Given a 208Y/120 power source, the ideal buck-boost transformer would
have 120 volts input and 18.56406460551 volts out. With the output
wired to boost the supply voltage, you get 138.56406460551 volts L-N
and thus 240 volts L-L. Such transformers are not actually made that
I can find.

My suggestion is to use a 240/120 to 24/12 transformer wired for 240
volts input and 24 volts output. The primary would be connected L-L
output would then be 20.8 (20.7846096908265) volts. This output
would then be wired in series with a particular L-N connections so
that it's phase angle is 30 degrees. Assuming a supply labeled as:

A
\
N---C
/
B

One B/B transformer fed from an A-B connection would have its output
at the polarity that corresponds to the A input connected to line B.
Showing this accurately is beyond ASCII graphics so I'll skip that
to avoid introducing new confusion. If this is wired as intended the
result is 120 volts at 0 degrees plus 20.7846096908265 volts at 30
degrees. This results in 138.390751136 volts at 4.306619096 degrees.
Do the same on all 3 phases with 3 such transformers. The resultant
L-L voltage is then 239.7 (239.699812265) volts, very close to the
desired 240 volts. Normal voltage variations are greater than this.
Assuming a perfectly resistive load on the 239.7Y/138.39 side of this,
the supply would see a current angle displaced 4.3 degrees for a power

I have seen many buck/boost wiring diagrams. But of all those seen,
none has ever done any cross-phase connection, not even of this small
amount. So I have these questions:

1. Is this an unknown method?

2. Is this prohibited by code?

4. Would a power factor of 0.997 be a problem where loads often have
their own power factors much worse than this?

There is another method that is a little more complex which involves
phase B being boosted by have as much as above from a reduced A-B as
well as from a reduced C-B. The same transformers can be used, but
would have to be wired with 12 volt secondaries not connected together
at all (or double the number of transformers at half the capacity).
This would result in a final L-N voltage of 138.000, and a L-L voltage
of 239.023, which is still very close to the target. But it would have
a 0 degree voltage phase shift. If the phase shift is more of an issue
then this approach is an alternative.

Or the inherint phase shift of the first setup could be used opposite
to the shift of a reactive load and presumably reduce its effect.

My real interest is not so much in supplying three phase loads at 240
volts, but rather, supplying single phase loads at 240 volts. Three
phase loads are more likely to be designed with the expectation of
the supply voltage being 208Y/120. But single phase loads may not.
Given a three phase source, getting nearly 240 volts is not hard. But
a common problem is that residences in large buildings are often given
2 legs of a three phase system as "fake" single phase (where the legs
are really 2 phases at 120 degrees). Getting 240 volts out of this is
my real goal. One could get three phase back from this like:

A A *
\ \ / \
N ... N C
/ /
B B

although that would have some substantially reactive current angles
on both of the supplied phases for any load on the output C-N. The
first or second examples I gave above might be applied on top of this
three phase hack, but only to get one phase of 240 volts at about the
A-B phase angle. Still, this is a lot of transformers and wiring at
this point.

By skipping the three phase step, 1 or 2 B/B transformers could be
used on a 2 phase 120/208 volt system two different ways. Probably
the simplest is one transformer with 240 volts in and a pair of
16 volts secondaries out. Wire the input to the 208 volt A-B and
use each of the secondaries to separately boost out A and B. This
will get only 235.5589 volts total. One side effect of this is that
the phase angle of A and B with respect to the neutral will increase
somewhat from 120 degrees to 126 (126.008983198) degrees.

Note that the above would have 132 (132.181693135) volts L-N and would
not be suitable for any L-N loads expecting 120 volts, and hence cannot
be used for any 3-wire 120/240 volt loads. Such loads would probably
have to use a 120 in 120/240 out autotransformer (half capacity) or a
208 in 120/240 out isolation transformer (full capacity) to get all the
right voltages. And these more expensive transformers due to the higher
capacities involved.

2. ### Guest

| wrote:
|>the viability of using buck/boost transformers to get that 240 volts,
|>either for delta connected three phase loads, or for single phase
|
| 208->230 or 208->240 is one of the most common uses of such
| transformers.

I've never seen a buck boost wiring diagram (and there are a LOT of
them in many manufacturer catalogs and spec sheets) that shows how
to do this for 208->240.

|>Given a 208Y/120 power source, the ideal buck-boost transformer would
|>have 120 volts input and 18.56406460551 volts out. With the output
|>wired to boost the supply voltage, you get 138.56406460551 volts L-N
|>and thus 240 volts L-L. Such transformers are not actually made that
|>I can find.
|
| Well, I think you should have googled before you did all that
| calculation.

Why do you think I didn't? You found calculations for all this
somewhere that I couldn't?

| A 120/240 - 12/24 would be used to go from 208 -> 230.

Not what I'm interested in.

| For 240, then 120/240 - 16/32 will be what you want, or close enough
| to it for most all purposes.
|
| If you really need 11 decimal places of accuracy, then simple
| transformers aren't going to do it for you under any circumstances.

The accuracy is just point figuring for math purposes. I do expect to
get to 240 volts within a couple volts. If they made a buck/boost
transformer with 18/36 output, that would hit it without having to use
the crossed phase angles. But 16/32 doesn't accomplish it. There are
no 18/36 volt out transformers from Google searches, nor any diagrams
showing how to get the correct voltage from 12/24 or 24/48 volt ones.

With 208, a pair of 16 volt boosts could get within 5 volts. Since they
don't make these as 18/36 volt transformers, I presume such a voltage
has other uses and getting 240 from 208 is not really considered in the
design.

3. ### Guest

The biggest variable will be what the PoCo sends you.
I consistantly see 247-248 on my single phase service but I have seen
it as low as 225 in my travels around town.

4. ### Guest

On Wed, 29 Nov 2006 20:42:17 -0500 wrote:

| On 29 Nov 2006 23:19:58 GMT, wrote:
|
|>I do expect to
|>get to 240 volts within a couple volts
| The biggest variable will be what the PoCo sends you.
| I consistantly see 247-248 on my single phase service but I have seen
| it as low as 225 in my travels around town.

How many of those 225's are really 208's a bit hot?

5. ### Don KellyGuest

Get rid of all the extra decimal places that your calculator is giving you-
they are actually meaningless as 120V (implied +/- 0.5V is not the same as
120.0000000V (implied +/-0.0000005 V)
If your calculator cannot be set to give a specific number of significant
figures, then I can sell you a simple fix consisting of a piece of masking
tape.
Noting that your 120V is likely to be in the range 114-126V normally- then
what is the point?
However, if the secondary windings of your buck boost transformer are
completely isolated from ground or the primary (and each other) then you can
do this. It's not a big deal.
As for phase shift- then consider phase shift with respect to what? Your
load sees a 3 phase voltage and acts accordingly. The primary voltage may be
shifted in phase but so is the primary current so that with a 1.0 pf load,
the primary sees a 1.0 pf load (ignoring reactive losses of the transformer-
where do you get a reactive input with no reactive output?). --

Don Kelly
----------------------------

6. ### Guest

| Get rid of all the extra decimal places that your calculator is giving you-
| they are actually meaningless as 120V (implied +/- 0.5V is not the same as
| 120.0000000V (implied +/-0.0000005 V)

I use the extra digits for a purpose. First, it ensures that accumulated
errors do not happen. Secondly, it helps determine if numbers are "matches"
that suggest simplified formulas. I've seen cases where NOT having them
can lead to suggested matches that really aren't just because certain paths
of arithmetic come close but don't hit the mark. For example I ran into
these numbers:
138.390751136
138.564064606
They look close. But they have different meaning. They just happen to be
close. But there is no mathematical equivalency to how they were calculated.
But if some calculation had come up with:
138.564064605
then the chance that it would be equivalent to the 2nd of the first 2 is
very high. The use of irrational numbers and a variety of different formulas
can result in "number crossings" that may or may not have significance.

| If your calculator cannot be set to give a specific number of significant
| figures, then I can sell you a simple fix consisting of a piece of masking
| tape.

I wouldn't use it.

When I see a reference to "208v" I figure it is probably the voltage involved
line to line when the line to neutral voltage is "120v". The confidence is
high, but it is not a certainty. But if I see "207.846v" then the confidence
is way up. Even more so with "207.84609690826527522329356v".

Yes I understand that voltage can vary. And I understand my arithmetic is
more accurate than the voltage.

| Noting that your 120V is likely to be in the range 114-126V normally- then
| what is the point?

The point is correct numeric identification. If the line to neutral voltage
can vary from 114 volts to 126 volts, then the line to line voltage can vary
from 197.45379206285201146 volts to 218.23840175367853898 volts

If I see "197v" somewhere, I don't know if it comes from just dropping "200v"
down by 3 volts or what. But if I see "197.45379v" then I have a very high
confidence that it was derived from 114 times ths square root of three.

| However, if the secondary windings of your buck boost transformer are
| completely isolated from ground or the primary (and each other) then you can
| do this. It's not a big deal.

I know how it's done.

| As for phase shift- then consider phase shift with respect to what? Your
| load sees a 3 phase voltage and acts accordingly. The primary voltage may be
| shifted in phase but so is the primary current so that with a 1.0 pf load,
| the primary sees a 1.0 pf load (ignoring reactive losses of the transformer-
| where do you get a reactive input with no reactive output?). --

You're saying that a transformer can store energy and have current flowing
in a secondary winding when the primary is at a zero crossing?

7. ### Bud--Guest

Use a 16/32V buck/boost. Connect the 32V boost from B to B'. Connect the
primary from A to B' - 208+32=240 applied to primary. The primary is
supplied from the boost side so the secondary supplies the primary

The 16V secondaries could be wired to boost A to A' and B to B' but I

A 24V boost from 208V to 232V is probably close enough.

Autotransformer conversion between 208 and 240 is specifically permitted
by the US-NEC 210.9-ex1 and 215.11-ex1

8. ### Ben MillerGuest

I was going to point him to exhibit 210.19 in the 2005 handbook, which shows
autotransformer connections both buck and boost between 208 & 240.

Ben Miller

9. ### Matthew BeasleyGuest

3 phase phase shifting transformers shift the phase of the voltatge and
current equally.

In your second proposed connection, the voltage was boosted out of phase.
The current in the primary consists of a current phasor in the shunt winding
added to the current phasor going through the series winding and the load.
If you apply a 1 PF load, the primary current will be zero when the primary
voltage is zero, negleticing transformer magnitizing current. The load
voltage won't be zero and the load current won't be zero at the same point
in time, but the current in the load and series winding will be canceled by
the current in the shunt winding (remember it's boosting another phase).

10. ### Guest

| wrote:
|>
|> | wrote:
|> |>the viability of using buck/boost transformers to get that 240 volts,
|> |>either for delta connected three phase loads, or for single phase
|> |
|> | 208->230 or 208->240 is one of the most common uses of such
|> | transformers.
|>
|> I've never seen a buck boost wiring diagram (and there are a LOT of
|> them in many manufacturer catalogs and spec sheets) that shows how
|> to do this for 208->240.
|>
| Use a 16/32V buck/boost. Connect the 32V boost from B to B'. Connect the
| primary from A to B' - 208+32=240 applied to primary. The primary is
| supplied from the boost side so the secondary supplies the primary

Sounds good. I haven't seen this one, but I should have thought of it.

My calculation tells me that if you start from 120 volts L-N dead on (for
reference purposes) and boost the 120 volt leg that is at 120 degrees
(also dead on phase angle) the boosted phase ends up at 148.555171749
volts at 113.821609305 degrees. The other phase(s) remain the same.
The L-L` ends up at 239.822419510 volts. That's close enough. But it
can only be used on appliances that are listed as accepting L-N voltage
at least at 148.555171749 nominal (more likely having to be around 160).

| The 16V secondaries could be wired to boost A to A' and B to B' but I

This ends up more balanced, though not perfectly so (because the angle
is not 180 degrees). Now you have L`-N at 134.084668120 for both phases
with angles of 116.582018908 and 243.417981092 (based on original phase
angles of 120 and 240). And you get the same exact 239.822419510 volts
as above, but at a slightly lower L`-N voltage.

I would rather wire it this way over the above way.

Note the precision calculations are based on 120.000000000 exact L-N
voltage.

It would be at least semi-balanced that way. Not well enough for powering
audio equipment, but probably OK for other things that don't need neutral.
The audio equipment could then be powered from a 1:1 isolation transformer
that would line up the phases at 180 degrees, or drop back down to running
the audio equipment at 120 volts with a 60-0-60 split.

| A 24V boost from 208V to 232V is probably close enough.

Except during brownouts. Consider the range of voltage you might get
and the range of voltage the appliance can work with. They should fit
if they are aligned right on center. But if you shift the supplied
voltage range down, the supply range may fall outside the utilization
range.

| Autotransformer conversion between 208 and 240 is specifically permitted
| by the US-NEC 210.9-ex1 and 215.11-ex1

In any configuration? What about three phase conversions? All diagrams
I have ever seen involve extending a L-N at the same exact angle it starts
at. And what if the source supply is closed delta (that probably violates
that rule I didn't look up, yet, because the only way to do that on a
corner grounded delta would end up with a tap-grounded delta with an odd
tap position for the grounding).

11. ### Guest

| writes:
|
|
|>| Get rid of all the extra decimal places that your calculator is giving you-
|>| they are actually meaningless as 120V (implied +/- 0.5V is not the same as
|>| 120.0000000V (implied +/-0.0000005 V)
|
|>I use the extra digits for a purpose. First, it ensures that accumulated
|>errors do not happen.
|
| The errors are already there. 120V +/- 0.5V means the voltage could be
| 120.0, 120.2, 119.5, 119.7, or an infinite number of other values between
| 119.5 and 120.5 volts. See below.

Of course. The reality is the supply is a statistical range. But it does
have a center. A given service drop may have a long term center that is
offset from 120 maybe even by quite a few volts. But the _basis_ voltage
is still 120 volts. I could run all my calculations on various voltages
in the range, and the extremes of the range, and get range results with
precision.

|>When I see a reference to "208v" I figure it is probably the voltage involved
|>line to line when the line to neutral voltage is "120v". The confidence is
|>high, but it is not a certainty. But if I see "207.846v" then the confidence
|>is way up. Even more so with "207.84609690826527522329356v".
|
| They teach the opposite in introduction to engineering courses. They
| teach the concept of "sig figs", significant figures. Simply put, you
| never use more digits after the decimal point than the accuracy you know,
| and when performing mathematical operations, you use the accuracy of the
| least known source and you must avoid adding more decimal places just
| because the calculator shows more digits. The extra decimal places in
| something like "207.84609690826527522329356 V" gives a false sense of
| accuracy.

Only to people that misunderstand the difference between precision and
accuracy. But then, I've found a lot of people that misunderstand that.
But such statistics won't change my way of doing things. If someone
misunderstands that I'm giving precision rather than accuracy, then I
say that is their problem, not mine.

Of course people don't care if it's 208.000000000 volts or 207.846096908
volts if they get shocked by it. Even a damned good meter isn't going
to give you all that many digits (a damned good one might do 207.846 and
a half-way decent one could do 207.8).

| As an example, how should an engineer write the line-line voltage of a 3
| phase system is the line-neutral voltage is given as 120. volts? The
| decimal states the "120" is known to 3 significant figures (voltage is
| within +/- 0.5 volts of 120 volts), and the square root of 3 is
| essentially known to a near infinite number of digits, the smaller of the
| two is 3 digits, so the correct answer is 208. volts. Same question, but
| if the voltage is given as 120.0 volts, the correct answer is 207.8 volts.
| If the line-neutral voltage is known only within +/- 5 volts, it should be
| written as "120" (no decimal point), as it's known to only 2 figures, and
| the line-line voltage should *not* be written as 208 volts (3 figures),
| but 210 volts (2 figures, rounded). Since the nominal "120 volts" is not
| supplied by the power company within 3 significant figures, all the "208
| volts" figures used everywhere is, from an engineering viewpoint, wrong!

First of all, saying "120" does NOT say that it is 3 significant figures.
It could be a reference label to the connection that is defined nominally
as 120 volts can could be found to be anywhere from 114 to 126 volts in
places where the voltage is within 5%, or even further at 108 to 132 volts
when 10% is the margin. And you can find different ranges based on long
pr short term statistics.

Nevertheless, I run my calculations at precision (not accuracy) based on
the "definition voltage". This lets me double check things, and assures
that where multiple steps should end up back at a whole number they very
likly will, or will be so damned close it's obvious to anyone. If you
do some calculations and the end result is 119.999999997 you can just
tell it was "meant to be" 120. But if the result is 119, that's not so
obvious.

My vector calculator uses long double precision, and rounds the results
to 9 fractional digits. I could recode it for more if I ever need to.
I also know the programming techniques to maximally preserve precision
in floating point calculations on a computer.

| This becomes ambiguous when the number of sig figs is less than the number
| of digits left of the decimal point (what does 69,000 volts mean?),
| but that's one of the reasons for scientific notation as well as metric
| prefixes. 69 kV has 2 figures accuracy, 69.0 kV 3 figures, 69.00 kV four
| figures etc.

I really don't know their basis voltage. If that is the L-N voltage on
a system with a L-L basis voltage of 120 kV, then the L-N _basis_ voltage
would be 69282.032302755 ... rounded to 9 fractional digits That
would be the center point in a range.

|>Yes I understand that voltage can vary. And I understand my arithmetic is
|>more accurate than the voltage.
|
| And when designing something, you must be sure the added digits don't come
| back and bite you.

Don't worry. I know how to handle high precision arithmetic.

|>If I see "197v" somewhere, I don't know if it comes from just dropping "200v"
|>down by 3 volts or what. But if I see "197.45379v" then I have a very high
|>confidence that it was derived from 114 times ths square root of three.
|
| When designing something, you should never have to guess the origin of a
| calculation as well as not being misled by the accuracy. A figure
| like "197.45379 +/- 0.5 volts" is absurd.
|
| And this concludes today's Engineering 101 class

When I was in school in the 1970's, courses in sciences and engineering
never explained anything about precision vs. accuracy. I found a lot of
people, including professors, didn't understand the distinction. More
recently I was reading a (rather heavy) college freshman chemistry book
(a chemistry prof my dad knew gave to him) and it actually did address
the precision vs. accuracy issue.

So would you change "197.45379 +/- 0.5" to be "197 +/- 0.5" or would you
change it to be "197.5 +/- 0.5"? Is 196.9530 within the range? What about
197.9536?

What you are doing when you change it to either 197 or 197.5 is called
quantization. It really is possible to have more precision than accuracy.
You could have a 6 digits voltmeter that reads 207.846 volts. Then you
can watch it waver up and down between 206.846 and 208.846. The accuracy
of the supply is +/- 1 volt while the precision of the meter _may_ be
+/- less than 0.001 volt. If the meter is calibrated to 0.0005 volts
accuracy, then its precision is easy to deal with. However, if it is
out of calibration, it can still give you precision. If you have a
reference voltage you can check the error. But even if it is out of
calibration by an unknown voltage, you might still read a figure around
120 volts on a circuit that should have around 120 volts and know that
it is at least not too far off. Then with the precision and lack of
accuracy, you can still measure the _other_ phase and see _about_ how
many volts different they are. You cannot do that if you misunderstand
precision and assume it is always bound to accuracy (sometimes it is,
but not in this case).

BTW, the square root of three is approximately:

1.7320508075688772935274463415058723669428052538103806280558069794519330169088

but most electrical engineers will use 1.732 or even 1.73 because it is quite

1.73 squared is 2.9929
1.732 squared is 2.999824
1.732050807568877 squared is 2.999999999999998983191099041129

The later is how far I have memorized the square root of three. But I can
calculate it with a program I wrote to many million digits if you'd want it.
When I want to know things like "does this calculation involving the square
root of 3 produce a result that is supposed to be at a whole number", then
I intentionally use more precision to be sure of it.

And I can also give you some whole number fractions that get very close to
the square root of 3. Here are some examples:

2011930833870518011412817828051050497 / 1161588808526051807570761628582646656
3484766425578155422712284885747939968 / 2011930833870518011412817828051050497
5496697259448673434125102713798990465 / 3173519642396569818983579456633697153

You can verify these by squaring the numerator, squaring the denominator,
and multiplying the denominator squared by 3 and getting the numerator
squared within a digit or two.

Of course someone will just say why not use:

1732050807568877293527446341505872367 / 1000000000000000000000000000000000000

Try it and you will see that the three I gave above have double the precision
as this one (about equal to the total number of digits expressed). I can also
do this for PI:

8904500941523253328862487239770856471 / 2834390681219723641210580762604275775

which is double the precision of:

3141592653589793238462643383279502884 / 1000000000000000000000000000000000000

I guess you can call me the master of trivial precision.

12. ### Guest

|
| |
|> | As for phase shift- then consider phase shift with respect to what? Your
|> | load sees a 3 phase voltage and acts accordingly. The primary voltage
|> may be
|> | shifted in phase but so is the primary current so that with a 1.0 pf
|> | the primary sees a 1.0 pf load (ignoring reactive losses of the
|> transformer-
|> | where do you get a reactive input with no reactive output?). --
|>
|> You're saying that a transformer can store energy and have current flowing
|> in a secondary winding when the primary is at a zero crossing?
|>
|
| 3 phase phase shifting transformers shift the phase of the voltatge and
| current equally.

So if you have a non-reactive pf 1 load on L-N and another on L-L the
current on the primary side will be exactly the same?

13. ### Don KellyGuest

---------
the two (exact) integers 2 and 2 are identical but this doesn't mean that
the calculations involved to reach them are the same. The above argument
doesn't apply. If you want to eliminate round off errors in calculations
then store the numbers between calculations -surely you are not entering
numbers with so many digits- My calculator can display the number of sig
figs that I want but the arithmetic done is to whatever floating point
precision that is involve. That doesn't mean that the results are more
accurate.
---------
---------
Your confidence is misplaced in that it has no relationship to reality.
when you say 120V, you are saying that it is a voltage between 119.5V and
120.5 V That is 3 significant figures. It is not the same as 120.0000 V.
Similarly, even though your calculator multiplies this by root(3) and gives
207.8460969...., the result is accurate only to 3 significant figures or
208V +/_ 0.5V. The rest is a tribute only to the capability of your
calculator. This is something that engineering schools try to drum into
their students. Solutions to problems that show answers like 207.8460 ....
lose marks if the original data doesn't justify this accuracy.
In addition, how often does a nominal 120V actually appear at the outlet?
You are working with nominal values
In rare cases, such as ill conditioned matrices, there is a need for double
precision arithmetic.
------
----------
So what is the confidence in the 114?
-------
No. Suppose that you have your Zig-zag secondary with a resistive load so
that both voltage and current are shifted 30 degrees (in the same direction)
from the reference (say phase a primary -consider a star primary for
simplicity). Now, if the secondary current is reflected into the primary as
you indicate then there would be a problem - you have secondary windings
from each pair of phases tied in series. That correctly implies a common
current in these two windings. However there is another phase a winding in
series with a c phase winding and the current in this is actually lagging
by 30 degrees (accounting for winding polarity) Now, on the basis of
secondary currents reflecting back to the primary, this implies that the
primary current of phase a is both leading and lagging. It also implies
that all 3 phase currents are in phase - double contradiction.
An easier argument is that the transformer doesn't produce or absorb
(ideally) any power or reactive (conservation of energy)so that for a given
secondary P and Q out, there must be the same P and Q input. If the loads
are balanced then that would also mean that the pf of the input is the same
as the output.
The answer to this quandary is that one must look at ampere turns, not
amperes.

Check it out in the simpler case of a single phase transformer 1200/240 V
with a center tapped secondary. If you have a load of 5A, unity pf at 120V
on one side and a load of 10A at 0.8 pf lag on the other leg. what is the
primary current? It cannot be 0.5A ,1.0 pf and 1.0A at 0.8 pf.
Considering each secondary separately you would have a=Np/Ns =10
Np*Ip =Ns* (Is1 +Is2)= Ns* (14.3 @ -24.8 degrees) or 1.43 A at 0.91 pf lag
Check from power
Ps1 =120*5 =600 watts Ps2 =120*8 =960 watts Qs1 =0 Qs2=120*6 =720 Vars
total 1560 watts, 720VAR or 1720VA at 0.91 pf lag and a current of 1.43A

For the 3 phase scheme this is also true. The primary amp turns must
balance the amp-turns of the two secondary windings in each phase. The
currents in these windings are 60 degrees apart and the number of turns is
the same in each.
Then considering that one of the phase a secondaries has a current leading
by 30 degrees, the other phase a secondary has a current effectively lagging
by 30 degrees so the sum of the ampere turns seen by the primary is NI1
+NI2 =N(root(3)I at angle 0 so for a balanced secondary load at unity pf,
the primary current which must balance the total secondary ampere turns- not
the amperes in the secondary windings) is at unity pf.

14. ### Don KellyGuest

-------

But doesn't the fact that the "precision" is an artifact of the calculator
and is actually meaningless mean anything?
To rely on such "precision" which occurs because the calculator is a stupid
beast and does all its work to whatever level of precision it is designed to
use and will treat an entry of 120 as 120.000000....
The extra zeros exist because it is can't do otherwise as it doesn't have
intelligence to do so. The user should have this intelligence. There is no
problem with this as it does reduce computational round off to an
insignificant level but when returning to the real world, the user should
have the intelligence to correlate the numerical values to the real world.
This reliance is a product of misunderstanding, not a cure for it.

By the way, Michael is quite right - I was giving 120, with no decimal
point, a sig fig level of 3 by interpreting it as 120.

15. ### Guest

|>Of course. The reality is the supply is a statistical range. But it does
|>have a center. A given service drop may have a long term center that is
|>offset from 120 maybe even by quite a few volts. But the _basis_ voltage
|>is still 120 volts. I could run all my calculations on various voltages
|>in the range, and the extremes of the range, and get range results with
|>precision.
|
| Such precision isn't necessary in that case, unless you're pulling a stunt
| like wiring two big substation transformers to have a zero voltage
| difference between two terminals, and using your two hands to "measure"
| the difference. In that case the accuracy doesn't matter (you don't care
| if the transformers are putting out 69 kv or 68 kv) but you do want the
| difference to be very precisely 0.

You're using "precision" and "accuracy" interchangeably here.

|>Only to people that misunderstand the difference between precision and
|>accuracy.
|
| I know the difference perfectly well.

|>First of all, saying "120" does NOT say that it is 3 significant figures.
|
| Right. To an engineer it's ambiguous, but it implies accuracy to two
| figures. "120.", to an engineer, states 3 figures accuracy.

You're using "precision" and "accuracy" interchangeably here.

|>It could be a reference label to the connection that is defined nominally
|>as 120 volts can could be found to be anywhere from 114 to 126 volts in
|>places where the voltage is within 5%, or even further at 108 to 132 volts
|>when 10% is the margin. And you can find different ranges based on long
|>pr short term statistics.
|
| There are different usages to an engineer and, say an electrician or a
| lineman working on something. Even today, electricians often speak of
| "two twenty" although the nominal voltage has been 240V (or at least
| larger than 220V for decades)

Indeed. "two twenty" is a label, not a measure. So is "220V" and "240V"
and even "207.84609690826527522329356". As labels they imply certain
meanings which have nothing to do with precisions or even measures. The
label "207.84609690826527522329356" means "120 times the square root of 3
with an extremely high degree of confidence" as opposed to "some number
that just happens to come out really close" (as often does happen when
many formulas with irrational numbers are involved).

|>Nevertheless, I run my calculations at precision (not accuracy) based on
|>the "definition voltage". This lets me double check things, and assures
|>that where multiple steps should end up back at a whole number they very
|>likly will, or will be so damned close it's obvious to anyone. If you
|>do some calculations and the end result is 119.999999997 you can just
|>tell it was "meant to be" 120. But if the result is 119, that's not so
|>obvious.
|

Tell that to the calculator or computer giving the results.

|>| This becomes ambiguous when the number of sig figs is less than the number
|>| of digits left of the decimal point (what does 69,000 volts mean?),
|>| but that's one of the reasons for scientific notation as well as metric
|>| prefixes. 69 kV has 2 figures accuracy, 69.0 kV 3 figures, 69.00 kV four
|>| figures etc.
|
|>I really don't know their basis voltage. If that is the L-N voltage on
|>a system with a L-L basis voltage of 120 kV, then the L-N _basis_ voltage
|>would be 69282.032302755 ... rounded to 9 fractional digits That
|>would be the center point in a range.
|
| Specifying "69282.032302755" as a center point in a range that can be a
| few hundred volts is absurd.

That depends on the context. If I want to specifically make a reference
to a baseline value related by a specific irrational number, more precision
increases the confidence of correctly matching the numbers. If I just want
to refer to a "69kv" circuit to by common label, to distinguish it from a
"12kv" circuit, those terms are enough. But if I do some arithmetic and what
I end up with is 68971.54879012, even though all steps used many digits of
precision, then I know one of two things: either an error was made somewhere
in the arithmetic, or the result happens to be close to, but is not the same
as, "69282.032302755". If I express either one to 2 digits, each would be
"69k". Precision in the arithmetic is more than just getting all the digits
(even though so many are not actually needed for use), but it is about
understanding what the number is. ONE of those numbers relates to exactly
120000 as a definition point by a ratio of the square root of three, whereas
the other does not. When doing arithmetic, this is important to perceive,
especially if you may be dealing with formulas that come full circle back
to numbers you already work with.

|>| And when designing something, you must be sure the added digits don't come
|>| back and bite you.
|
|>Don't worry. I know how to handle high precision arithmetic.
|
| YOU know, but does anyone using your design documents, should you be
| designing something?

Lots of people certainly don't know how to deal with precision. Lots of
people say "accuracy" when precision is what is meant ... or visa-versa.

And who says I would put that on a design document. I would where I feel
it is important to identify that meaning of a number. But where meaning
is unimportant and construction value is all that is needed, I would supply
the value and its tolerance.

BTW, everything would all be different if we used base 8 or base 12 instead
of base 10. We "quantize" to base 10 when we round things to make values
fit available accuracy. Then we force precision to that level. It would
end up at different points in a different arithmetic base. Thus we can
actually have differences in engineering or construction that relate to
what base we use.

| If you needed to specify the rating of a capacitor to connect line-line
| in a 120 volt line-neutral 3 phase circuit, would you specify a capacitor
| with a breakdown voltage of 293.9387692 volts (120V * sqrt(3) * sqrt(2)
| sine wave peak voltage, plus a hundred picovolts safety margin) ?

293.93876913398137178367408896470696703591369767880041541192310807

Seriously, it depends on the context. If I'm trying to match numbers,
I'd use more digits just to be sure the matching confidence is high.
Normally, for a capacitor, I'd want a lot more margin to account for
variation in voltage, environment, manufacturing, and aging. So I am
likely to say "at least 600 volts" and may even go higher.

100 years ago you could be a "1000 pound rope" and never see it break
even with 2000 pounds of load on it. Today, you buy a "1000 pound rope"
and you can't sue the manufacturer for it breaking under your 1005 pound
load. That's a misuse of the numbers. And it certainly is not precision.
It's just being cheap.

|>When I was in school in the 1970's, courses in sciences and engineering
|>never explained anything about precision vs. accuracy. I found a lot of
|>people, including professors, didn't understand the distinction. More
|>recently I was reading a (rather heavy) college freshman chemistry book
|>(a chemistry prof my dad knew gave to him) and it actually did address
|>the precision vs. accuracy issue.
|
| This is an accuracy issue, but it fails on precision grounds as well.
| Are the three transformers in a 3 phase circuit _precisely_ identical
| to that many degrees precision? Is the phase angle _precisely_ 120.000000
| degrees? Is the resistance/capacitance/indeuctance of the 3 supplying
| conductors _precisely_ the same? Etc.

This is called "tolerance".

Of course not.

But when one is discussing the bounds of theoretical and comparing results
to confirm formulas, the numbers have a different set of semantics.

|>So would you change "197.45379 +/- 0.5" to be "197 +/- 0.5" or would you
|>change it to be "197.5 +/- 0.5"? Is 196.9530 within the range? What about
|>197.9536?
|
| What is this "197.45379" figure? A one time measurement? A calculation?
| If a calculation, what is the accuracy of the figures used in the
| calculation? What is the "+/- 0.5" figure? A specification? An observed
| range? An error figure carried through the calculation (say double a
| figure of 98.726895 +/- 0.25) ? What is the accuracy of the "0.5" figure?
| Where did all those decimal points come from, was it multiplication or
| division of something with few significant figures by an irrational number
| such as sqrt(3)? Depending on the answer to these, it may be appropiate
| to write it as 197. or 197.5.

It also depends on whether you are talking about real physical things
that can be measured to a certain accuracy, or discussing theory and
the mathematics behind it.

| There are instances where you have to bring both a figure and the error
| specification through the calculation.
|
|>What you are doing when you change it to either 197 or 197.5 is called
|>quantization. It really is possible to have more precision than accuracy.
|>You could have a 6 digits voltmeter that reads 207.846 volts. Then you
|>can watch it waver up and down between 206.846 and 208.846. The accuracy
|>of the supply is +/- 1 volt while the precision of the meter _may_ be
|>+/- less than 0.001 volt. If the meter is calibrated to 0.0005 volts
|>accuracy, then its precision is easy to deal with. However, if it is
|>out of calibration, it can still give you precision. If you have a
|>reference voltage you can check the error. But even if it is out of
|>calibration by an unknown voltage, you might still read a figure around
|>120 volts on a circuit that should have around 120 volts and know that
|>it is at least not too far off. Then with the precision and lack of
|>accuracy, you can still measure the _other_ phase and see _about_ how
|>many volts different they are. You cannot do that if you misunderstand
|>precision and assume it is always bound to accuracy (sometimes it is,
|>but not in this case).
|
| Again, in many cases, you have to bring both the figure and the error
| range through the calculations. But unless the accuracy error somehow
| cancels (such as subtracting two figures known to have total inaccuracy
| from the same source) that extra precision is useless.

And where I need to bring the error range through, I sure will. And it
sometimes influences things in unexpected ways (especially with things
like non-linear effects). But that wasn't needed in my recent posts.

|>BTW, the square root of three is approximately:
|
|>1.7320508075688772935274463415058723669428052538103806280558069794519330169088
|
| Irrelevant to every engineering application I can think of.

However, keep in mind that is an irrational number. It is always in some
amount of error. Depending on what kind of math is involved, you might
need more precision for _this_ number and the calculations using it. For
example, if a calculation has to be repeatedly multiplied and divided by
this number as new factors keep being introduced, this error builds up.
More precision to match the scale of the arithmetic (e.g. how many times
a program loop doing this has to run).

Chaos theory describes how this affects calculations. It is also termed
the "butterfly effect". Was that storm that came through caused by a
butterfly somewhere? No. Could it have been influenced in some minute
way? Yes. Not in ways you could measure, but in ways much larger than
it's small size and mass would suggest.

|>but most electrical engineers will use 1.732 or even 1.73 because it is quite
|
| Exactly.

.... where precision isn't needed ... but engineers need to know where it is
not needed, and where it _is_ needed. I think too few can do this well.

|>The later is how far I have memorized the square root of three. But I can
|>calculate it with a program I wrote to many million digits if you'd want it.
|>When I want to know things like "does this calculation involving the square
|>root of 3 produce a result that is supposed to be at a whole number", then
|>I intentionally use more precision to be sure of it.
|
| Good engineering can justify calculations without that, plus anything
| beyond the simplest it really can't be done anyway.

|>And I can also give you some whole number fractions that get very close to
|>the square root of 3. Here are some examples:
|
| Useless except to a math geek.
|
| I can tell you for a fact that in introduction to engineering course,
| writing the line-line voltage of a 120.V line-neutral circuit as
| 207.8460969V will be marked wrong.

Most likely what is being asked for is a label, not mathematical matching.

What would you write for the line to line voltage for a circuit with 230V
as the line to neutral voltage? 398V? 400V?

Did you see my proposal for an electrical system (if I could go back in
time ... a hypothetical exercise that could not really happen today) where
I proposed a standard voltage of 288 volts line to line, and mentioned that
it would be based on 144 volts line to neutral from single phase sources
and 166 volts line to neutral from three phase sources. Then I went on to
indicate that a higher voltage system could use that same 288 volts in a
line to neutral way and have 499 volts line to line. I mentioned that I
thought the voltages were "catchy" because they all ended in two like
digits. But I wonder if such a system have been used, how many people
would still end uo just calling the latter "500V".

Humans LOVE round numbers. I think this is excessively so. So did you
pick 398V or 400V above? I bet it was 400V. Nice round number.

16. ### Guest

| |
| But doesn't the fact that the "precision" is an artifact of the calculator
| and is actually meaningless mean anything?

If the precision is so meaningless, what would you do if each time you
did use a calculator, those "extra meaningless digits" were randomly
different?

| To rely on such "precision" which occurs because the calculator is a stupid
| beast and does all its work to whatever level of precision it is designed to
| use and will treat an entry of 120 as 120.000000....

Wo what would you do if each time you used a calculator that always gave
you 16 digits of precision was also giving you random variations in the
lowest 4 digits when you didn't need even anywhere near 12? If you press
"3" then "SQRT" and keep getting:

1.732050807566414
1.732050807569301
1.732050807569073
1.732050807568584
1.732050807568319
1.732050807565962
1.732050807569763
1.732050807567522
etc

Would you just ignore the last 4 digits as meaningless, or wonder what the
calculator was doing? All of these values are plenty precise for work in
three phase power systems, so why should anyone care?

| The extra zeros exist because it is can't do otherwise as it doesn't have
| intelligence to do so. The user should have this intelligence. There is no
| problem with this as it does reduce computational round off to an
| insignificant level but when returning to the real world, the user should
| have the intelligence to correlate the numerical values to the real world.
| This reliance is a product of misunderstanding, not a cure for it.

It depends on how the numbers are being used. If they are being used to
show what formula was being used to derive these numbers, precision can be,
and is, very useful. In most cases just a few more digits does the job.
I've seen cases where a precision of more than 20 digits was needed to
verify a correct formula.

17. ### Guest

|
|
| |>
|> | Get rid of all the extra decimal places that your calculator is giving
|> you-
|> | they are actually meaningless as 120V (implied +/- 0.5V is not the same
|> as
|> | 120.0000000V (implied +/-0.0000005 V)
|>
|> I use the extra digits for a purpose. First, it ensures that accumulated
|> errors do not happen. Secondly, it helps determine if numbers are
|> "matches"
|> that suggest simplified formulas. I've seen cases where NOT having them
|> can lead to suggested matches that really aren't just because certain
|> paths
|> of arithmetic come close but don't hit the mark. For example I ran into
|> these numbers:
|> 138.390751136
|> 138.564064606
|> They look close. But they have different meaning. They just happen to be
|> close. But there is no mathematical equivalency to how they were
|> calculated.
|> But if some calculation had come up with:
|> 138.564064605
|> then the chance that it would be equivalent to the 2nd of the first 2 is
|> very high. The use of irrational numbers and a variety of different
|> formulas
|> can result in "number crossings" that may or may not have significance.
| ---------
| the two (exact) integers 2 and 2 are identical but this doesn't mean that
| the calculations involved to reach them are the same. The above argument
| doesn't apply. If you want to eliminate round off errors in calculations
| then store the numbers between calculations -surely you are not entering
| numbers with so many digits- My calculator can display the number of sig
| figs that I want but the arithmetic done is to whatever floating point
| precision that is involve. That doesn't mean that the results are more
| accurate.

The calculations _may_ be equivalent. Or they may not be. But I know from
experience that when irrational numbers are involved (like most square roots)
the chance of a false positive prefect match is very low.

|> When I see a reference to "208v" I figure it is probably the voltage
|> involved
|> line to line when the line to neutral voltage is "120v". The confidence
|> is
|> high, but it is not a certainty. But if I see "207.846v" then the
|> confidence
|> is way up. Even more so with "207.84609690826527522329356v".
| ---------
| Your confidence is misplaced in that it has no relationship to reality.
| when you say 120V, you are saying that it is a voltage between 119.5V and
| 120.5 V That is 3 significant figures. It is not the same as 120.0000 V.
| Similarly, even though your calculator multiplies this by root(3) and gives
| 207.8460969...., the result is accurate only to 3 significant figures or
| 208V +/_ 0.5V. The rest is a tribute only to the capability of your
| calculator. This is something that engineering schools try to drum into
| their students. Solutions to problems that show answers like 207.8460 ....
| lose marks if the original data doesn't justify this accuracy.
| In addition, how often does a nominal 120V actually appear at the outlet?
| You are working with nominal values
| In rare cases, such as ill conditioned matrices, there is a need for double
| precision arithmetic.

When you say that "120V" means a voltage between "119.5V" and "120.5V"
(which I don't necessarily agree with), what are you saying when you say
"119.5V"? Are you saying "a voltage between 119.45V and 119.55V" ??

Range has to be defined. It is so often _defaulted_ to be defined as
have a precision step above or below. But that is not always reality.
A range might be 3/4 of a precision step above or below.

"120V" might be a range from "114V and 126V".

| ------
|>
|> Yes I understand that voltage can vary. And I understand my arithmetic is
|> more accurate than the voltage.
|>
|>
|> | Noting that your 120V is likely to be in the range 114-126V normally-
|> then
|> | what is the point?
|>
|> The point is correct numeric identification. If the line to neutral
|> voltage
|> can vary from 114 volts to 126 volts, then the line to line voltage can
|> vary
|> from 197.45379206285201146 volts to 218.23840175367853898 volts
|>
|> If I see "197v" somewhere, I don't know if it comes from just dropping
|> "200v"
|> down by 3 volts or what. But if I see "197.45379v" then I have a very
|> high
|> confidence that it was derived from 114 times ths square root of three.
| ----------
| So what is the confidence in the 114?

That depends on what I'm relating the confidence to. There would be a
high level of confidence if I am relating 114 to 197.45379 divided by
the square root of three. But it certainly is possible for the 114 to

| No. Suppose that you have your Zig-zag secondary with a resistive load so
| that both voltage and current are shifted 30 degrees (in the same direction)
| from the reference (say phase a primary -consider a star primary for
| simplicity). Now, if the secondary current is reflected into the primary as
| you indicate then there would be a problem - you have secondary windings
| from each pair of phases tied in series. That correctly implies a common
| current in these two windings. However there is another phase a winding in
| series with a c phase winding and the current in this is actually lagging
| by 30 degrees (accounting for winding polarity) Now, on the basis of
| secondary currents reflecting back to the primary, this implies that the
| primary current of phase a is both leading and lagging. It also implies
| that all 3 phase currents are in phase - double contradiction.
| An easier argument is that the transformer doesn't produce or absorb
| (ideally) any power or reactive (conservation of energy)so that for a given
| secondary P and Q out, there must be the same P and Q input. If the loads
| are balanced then that would also mean that the pf of the input is the same
| as the output.
| The answer to this quandary is that one must look at ampere turns, not
| amperes.
|
| Check it out in the simpler case of a single phase transformer 1200/240 V
| with a center tapped secondary. If you have a load of 5A, unity pf at 120V
| on one side and a load of 10A at 0.8 pf lag on the other leg. what is the
| primary current? It cannot be 0.5A ,1.0 pf and 1.0A at 0.8 pf.
| Considering each secondary separately you would have a=Np/Ns =10
| Np*Ip =Ns* (Is1 +Is2)= Ns* (14.3 @ -24.8 degrees) or 1.43 A at 0.91 pf lag
| Check from power
| Ps1 =120*5 =600 watts Ps2 =120*8 =960 watts Qs1 =0 Qs2=120*6 =720 Vars
| total 1560 watts, 720VAR or 1720VA at 0.91 pf lag and a current of 1.43A
|
| For the 3 phase scheme this is also true. The primary amp turns must
| balance the amp-turns of the two secondary windings in each phase. The
| currents in these windings are 60 degrees apart and the number of turns is
| the same in each.
| Then considering that one of the phase a secondaries has a current leading
| by 30 degrees, the other phase a secondary has a current effectively lagging
| by 30 degrees so the sum of the ampere turns seen by the primary is NI1
| +NI2 =N(root(3)I at angle 0 so for a balanced secondary load at unity pf,
| the primary current which must balance the total secondary ampere turns- not
| the amperes in the secondary windings) is at unity pf.

I see now we weren't taking about the same thing. I was only looking at
when the same thing is down in a balanced three phase system.

Consider the typical "2 legs of 208Y/120" service where an autotransformer
derives the missing leg:

A Y A Y
\ -> \ \ / \
N Z Y ==> N Z
/ ------> / /
B N B

This is certainly not very well balanced.

Now you get a voltage between X and Z at a phase angle of the phase that
is missing. Put a load between X and Z. What phase angle do you get on
the incoming A-N and B-N? What pf? 0.5? Leading or lagging?

I could also do:

A Y A Y
\ -> \ / \ / \
N Z Y A ==> W N Z
/ ------> / --> /
B N W

Hey, now I have the 240 volts I wanted, and with a neutral right in between.
Of course I could have gotten it from either 120 volt leg with a 1:2 step up.
But at least I am putting half the current on A and half on B. It just has
a lousy power factor.