What is the resistance of a cube

[Kilowatt]

What is the resistance of a cube that has a 1 ohm resister on each side
measured between opposite corners?

 

[krw]

What is the resistance of a cube that has a 1 ohm resister on each side
measured between opposite corners?

Flatten out the cube. Assume your two corners are 'X' and 'Y', and the
corners that are attached to 'X' through one resistors are 'A', 'B',
and 'C'. The corners thusly connected to 'Y' are 'D', 'E', and 'F'.
You then have three resistors XA, XB, and XC and three symmetrical at
the other corner: YD, YE, and YF. Inbetween there are six resistors;
AD, AE, BD, BF, CE, and CF. By symmetry we can see that the voltage at
A=B=C and D=E=F, so we can consider all the resistors in these three
sets in parallel. Thus the total resistance is (XA||XB||XC)+
(AD||AE||BD||BF||CE||CF)+(YD||YE||YF), or 1/3 + 1/6 + 1/3.

 

[John McGaw]

What is the resistance of a cube that has a 1 ohm resister on each side
measured between opposite corners?

Gawd! Is that one still floating around? That was one of the questions given
in the early stages of the basic electronics course in my junior year of HS
back in 1963...

 

[Jimmie]

Gawd! Is that one still floating around? That was one of the questions
given
in the early stages of the basic electronics course in my junior year of
HS
back in 1963...

Seems it never goes away. When I first saw it in the late 60's it was
presented to me and a friend of mine by a retired engineer from GE. My
friend who was in college at the time took about 2 days and several sheets
of paper to work it out. I was a junior in HS and worked it out in my head
in about fifteen minutes. Not bragging, I saw the symetry and the easy way
to work it. My friend attacked it in a way that did not depend on the
symetry of the network.I am sure his better understanding of networks led
him in the direction he went and my ignorance in mine.

 

[Sylvia Else]

Build the cube and drive with a 3 amp ideal current source.
Apply Kirchoff's Current Law in presence of symmetry to see how
the current splits evenly along edges of the cube. Then add up
the voltage drops along any 1 path, divide by 3 amps, and
you have the resistance.

Now a more interesting question is replace the 1 ohm resistors
with 470-ohm, silver fourth band (10% tolerance) resistors.
How does the "tolerance" of this effective network compare to
the 10% tolerance of the individual components? For simplicity,
assume that the individual resistors fall within a uniform
distribution from 423 through 517 ohm.

Why?

When considering tolerances, one isn't interested in a design that will
probably work given the distribution of component values within their
tolerance limits. One wants a design that definitely will work even if
all the components are at their worst case permitted values.

So the tolerance of the cube made of 10% resistors (of the same nominal
value) is still 10% isn't it?

Sylvia.

 

[daestrom]

Why?

When considering tolerances, one isn't interested in a design that will
probably work given the distribution of component values within their
tolerance limits. One wants a design that definitely will work even if all
the components are at their worst case permitted values.

So the tolerance of the cube made of 10% resistors (of the same nominal
value) is still 10% isn't it?

But I think his point is more about how 'balanced' it will be. Presumably
the cube is used instead of a single resistor for a reason. If some
resistors are high, while others low, how will the imbalance affect some of
the current distribution. Will the non-uniformity upset the current balance
enough to dissipate too much power in some resistors?

Admittedly, it is a 'contrived' problem, but it does illustrate that "things
are not always what they seem..."

daestrom

 

[Sylvia Else]

Part of any design is understanding the variability in its
characteristics and how that variability is a function
of all the individual circuit components' variabilities.
There are lots of reasons for doing this. One is to compare
the robustness of two designs -- hopefully choosing the one
that is less susceptible to individual component variability.
Another area is in diagnosing and localizing circuit faults.

No, sorry, I don't get this. If a design might not work properly when
some components are at the limit of their tolerance, then the design
should specify components with closer tolerances.

 

[Sylvia Else]

What's to not get? All designs are subject to the robustness
of their constituent components. It is this up-front analysis
that actually leads to the conclusion of that certain tolerance
value resistors are sufficient to get the job done. Imagine
if you are presented with a brittle design that works only when
all components are within 0.00000001% tolerance, then the design
is unrealistic, costly, and plain bad.

I do not dispute that, but I cannot see its relevance.

What I should probably have asked is what you actually mean by the
"tolerance" of the effective network.

When we say that a resistor has a tolerance of 10%, we mean that its
true value may differ from its nominal value by up to 10% (of its
nominal value). The obvious meaning of the tolerance of the network is
therefore the extent by which its true value may differ from the value
calculated using the nominal value of its resistors. You introduced the
issue of the distribution of values, but there's no way of using that
concept in calculating the tolerance of the network according to the
usual meaning, so you presumably meant something else.

What?

Sylvia.

 

[daestrom]

No, sorry, I don't get this. If a design might not work properly when some
components are at the limit of their tolerance, then the design should
specify components with closer tolerances.

Yes, but how do you know, "If a design might not work properly when ...."?

You have to do the analysis with the various components at their limits and
understand what happens if a certain one is at its high limit while another
is at its low limit. That's all, just saying such analysis is an
'interesting' problem.

If it turns out that it won't work under such circumstances, then you're
absolutely right. Time to change the design and/or specs. But, "How do you
know?"

daestrom

 

[Sylvia Else]

Yes, but how do you know, "If a design might not work properly when ...."?

You have to do the analysis with the various components at their limits and
understand what happens if a certain one is at its high limit while another
is at its low limit. That's all, just saying such analysis is an
'interesting' problem.

If it turns out that it won't work under such circumstances, then you're
absolutely right. Time to change the design and/or specs. But, "How do you
know?"

I have no disagreement with that, but that's not what the proposed
modified question required. There was talk about a distribution of
values, not worst case scenarios.

Sylvia.

 

[daestrom]

I have no disagreement with that, but that's not what the proposed
modified question required. There was talk about a distribution of values,
not worst case scenarios.

Perhaps I was reading it differently. I'm thinking more of 'works' includes
getting a certain voltage division between vertices, or the total resistance
is 5/6 ohm with a certain confidence when the thing is constructed of
individual parts whose tolerances are spread with a certain distribution.
Kind of like making them on an assembly line and you want to know how many
will pass a test that measures the voltage between two vertices. Using just
standard 10% components, if the distribution of values is 'normal', or
'flat', or'chi', or whatever, what would be the variation in performance.
(i.e. how many will fail the QC test and have to be re-worked).

After all, if building up a resistor network to make one 'composed resistor'
with 10% tolerance parts, given enough individual parts, the variations
*should* cancel out in the overall system and give you a 'composed resistor'
with even better tolerance than the parts. And how will the variation of
such a resistor network be distributed compared the distribution of
variation in the individual components.

daestrom

 

[Sylvia Else]

Perhaps I was reading it differently. I'm thinking more of 'works' includes
getting a certain voltage division between vertices, or the total resistance
is 5/6 ohm with a certain confidence when the thing is constructed of
individual parts whose tolerances are spread with a certain distribution.
Kind of like making them on an assembly line and you want to know how many
will pass a test that measures the voltage between two vertices. Using just
standard 10% components, if the distribution of values is 'normal', or
'flat', or'chi', or whatever, what would be the variation in performance.
(i.e. how many will fail the QC test and have to be re-worked).

After all, if building up a resistor network to make one 'composed resistor'
with 10% tolerance parts, given enough individual parts, the variations
*should* cancel out in the overall system and give you a 'composed resistor'
with even better tolerance than the parts. And how will the variation of
such a resistor network be distributed compared the distribution of
variation in the individual components.

Such calculations are certainly possible. I'd question whether it would
be a realistic approach to production.

As something of an aside, I think it used to be the case that resistors
were marked according to the actual tolerance they achieved. That is,
you made bunch of resistors, then measured them. The ones within 1% were
marked accordingly, and sold at a high price. Next came the 5% ones, at
a lower price, then the 10% ones, and finally, the 20% ones (no
tolerance band). So if I bought 10% resistors, the one thing I could be
sure about was that they were not within 5% of their nominal value.

I doubt it's still true, though.

Sylvia.

 

[Sylvia Else]

If they did that, why not just mark the measured value on them and sell
them as 1% (or .0001%, just need a better tester) tolerance resistors?

That's a good question.

No market? I suppose it's a form of yield management.

Those designs that require 1% resistors would mostly be because they
needed a consistency in value, wouldn't they, rather than because they
need to be able to choose a value? I don't remember seeing many very odd
valued 1% resistors. Typically, more like the values in the 5% range,
just with 1% tolerance.

I did once come across 1% resistors that had an extra band, and the next
lower multiplier. I don't know about others, but over time I've tended
to have the colour combination for common resistor values in my head, so
I recognise them directly, without having to decode them. The extra band
and lower multiplier was a damned nuisance.

Sylvia.

 

[daestrom]

Such calculations are certainly possible. I'd question whether it would be
a realistic approach to production.

As something of an aside, I think it used to be the case that resistors
were marked according to the actual tolerance they achieved. That is, you
made bunch of resistors, then measured them. The ones within 1% were
marked accordingly, and sold at a high price. Next came the 5% ones, at a
lower price, then the 10% ones, and finally, the 20% ones (no tolerance
band). So if I bought 10% resistors, the one thing I could be sure about
was that they were not within 5% of their nominal value.

I doubt it's still true, though.

Reminds me of the story about the old 'Double-Sided' vs 'Single-Sided'
floppy disks. Supposedly all manufactured the same way, those that passed
on both sides were labeled 'Double-Sided' and sold for a higher price than
those that failed one side and were thus labeled 'Single-Sided'.

daestrom

 

[Sylvia Else]

Reminds me of the story about the old 'Double-Sided' vs 'Single-Sided'
floppy disks. Supposedly all manufactured the same way, those that passed
on both sides were labeled 'Double-Sided' and sold for a higher price than
those that failed one side and were thus labeled 'Single-Sided'.

I think it is still true of CPUs. They're all made the same way, then
tested. The ones that perform properly at the higher clock rates are
then labelled that way, and sold at the higher prices.

BTW, who are the people who pay top dollar for today's top of the range
speed, that becomes tomorrows totally obsolete version?

Sylvia.

 

[Sylvia Else]

Not me, this is a P166 I am typing on

I've just decided upgrade from a P3 600 to an P3 866 which I've bought
second hand for $50. Mainly because the 600 only has a 100Mhz FSB, and I
need a memory upgrade (Java is so memory thirsty). Turns out that buying
a processor with a 133Mhz FSB means I save on the memory upgrade -
100Mhz memory is difficult to get now, and priced accordingly.

No doubt someone will tell me I could have underclocked the 600 clock,
but overclocked its FSB. I wasn't sure that that would work, of if it
did, that it'd be reliable.

Not that any of you wanted to know that.

Sylvia.

 

[krw]

I think it is still true of CPUs. They're all made the same way, then
tested. The ones that perform properly at the higher clock rates are
then labelled that way, and sold at the higher prices.

It's more or less still true, though with more twists. One can tweak
the processing to produce higher yields at lower speeds, or more higher
speed units at a reduced yield. One can similarly tweak the process to
optimize for power and speed. If customers are ordering higher speed
parts than your process wants to make, turn the knobs to produce what
the customer wants. Productivity may suffer, but it beats having a lot
of parts that no one wants.

BTW, who are the people who pay top dollar for today's top of the range
speed, that becomes tomorrows totally obsolete version?

The same ones that will buy tomorrow's "top of the range" part. ;-)