Estimating Parallel Resistor Circuits
without a Calculator, and still have
less than 5% error !
Forward:
Here is an example of a procedure that may
seem at first a little bit awkward or confusing, and initially
learning to use that process may take just a little bit of practice.
Even though this process is for only 2 resistors at a time, any
experience at all will allow you to pair up 2 and then pair up
the result with another resistor ... and so on.
Bear in mind that it is the using
of a particular technique or that makes it worthwhile to
learn. You will find that all the "old-timers"
in this business, because of their long term experience, instinctively
use some of these methods without necessarily thinking about
the process they are using at the time. If you should ask them
how they do it, they may very well say "lots of experience",
rather than trying to explain exactly how.
Well, here's how you can learn in a short
while, what it took them otherwise years to accomplish. You will
be amazed with what just a little bit of practice with these
examples will allow you to do.
Method #1: I call this one the "Upper-Limit"
vs "Lower-Limit" method.
[ For 2 resistors of values that are less
than a 2:1 ratio to each other ]
Method #2: I call this one the "Ratio-Plus-1"
method.
[ For 2 resistors of values that are less
than a 10:1 ratio to each other ]
Method #3: I call this one the "Sliver" (or
very small slice) method.
[ For 2 resistors of values that are more
than a 10:1 ratio to each other ]
A special note:
These 3 methods may seem at first to be somewhat awkward,
but they are actually very effective. It is only because of the
necessary wording and lengthy explanations that are necessary
here, that may make them appear awkward. They do take just a
little bit of getting used to, but just a little bit of practice
will allow you to quickly and easily approximate resultant values
that are surprisingly close to what they should be.
"Upper-Limit" vs "Lower-Limit" method: [ For 2 resistors of values that are less than a 2:1
ratio to each other ]
{To practice this exercise, you need to draw a sketch of
2 resistors in parallel}
PDF
File (8Kb)
"Ratio + 1" method: [ For 2 resistors
of values that are greater than a 2:1 ratio, but less than a 10:1
ratio to each other ]
{To practice this exercise, you need to draw a sketch of
2 resistors in parallel}
This method is quite straight forward, in that you approximate
the ratio of the 2 resistors that are in parallel with each other,
and add 1 to that ratio.
PDF
File (5Kb)
Example: Take a 33K resistor and a 330K resistor, place them
in parallel with each other. It should be obvious that the ratio
between those 2 resistors is 10:1.
Now take that 10:1 ratio, add 1 to make it 11:1.
Divide the larger of the 2 resistor values (which is the
330K) by the 11 of the new ratio, and the result will be 30K
!!
Of course, with approximating the ratio of real resistor
values can be apparently a little sloppy, but you will be amazed
just how close you can be. Remember, we are only approximating
these results, but practice really helps.
Another example: Take a 3.9K and a 6.8K and place them in
parallel with each other. Since I personally don't like to even
try to figure really odd values, I simply consider the 3.9K as
almost 4K and the 6.8K as almost 7K. Now, that ratio might be
just a little less than a simple 2:1 ratio, so I'll call my "Ratio
+ 1" a little less than a 3:1.
Now, simply divide the larger of those 2 parallel resistors
(the 6.8K) by my <3, and approximate the result as approximately
2.4K.
Remember that when you divide a number by something "less
than", your result will be "more than", which
is why when we might have taken the 6.8K (think of the 6.8K as
6.9K for easy division) and divided it by less than the
3:1 ratio, we will get a number that is more than the
expected 2.3K. We can call it approximately 2.4K.
Now, if you calculate the actual result by conventional means,
the result will be found to be about 2.479K.
This means that we estimated an answer that turned out to
be 96.8% correct, and therefore only an error of 4.2% !!
This method does take a little more practice than the "Lower-Limit"
vs "Upper-Limit" method, to get a good handle on it,
but with practice..........
A special note: This method, done with a calculator, is a
3rd method not commonly given in the text books, but is accurate.
"Sliver" method: [ For 2 resistors
of values where one is greater than a 10:1 ratio to the other
]
{To practice this exercise, you need to draw a sketch of
2 resistors in parallel}
In presenting this method, I like to illustrate what I call
the difference between "Fudge-Factor" and "Finagle's-Constant",
where the "Fudge-Factor" is when you discover an error
in your calculations that can be brought into line by "Fudging"
your answer by just the right amount... Now, suppose that you
somehow knew ahead of time that your resultant answer was going
to be slightly wrong (and about how much), and you determined
that if you could "Finagle" the parameters ahead of
time, and then you wouldn't have to "Fudge" later. Hmmm....
First, we need to explain our "Finagle-Parameters":
It can be considered that an error caused by disregarding
a parallel resistor which has a value of 10 times more than the
first resistor (i.e. a 10:1 ratio) will cause a 10% error.
It can also be considered that by disregarding parallel resistors
of a 100:1 ratio will cause only a 1% error.
Finally, it can be considered that disregarding parallel
resistors of a 1000:1 will cause only a 0.1% error.
Here we can see that if we can estimate the amount of possible
error ahead of time, we can simply remove a "Sliver"
off of the lowest value, with the size of the "Sliver"
depending on the amount of expected error.
PDF
File (16Kb)
Example #1: Place a 27K in parallel with a 270K resistor.
Now if we disregarded the 270K resistor and called the result
27K, we would be in error because the result is always "less
than the least". But, if we knew that ahead of time, we
could simply take about a 10% slice ("Sliver") off
of the 27K (which is 2.7K), and we wind up with an approximate
value of 24.3K as a result. [24.54K actual, for 99% accuracy
]
Example #2: Place a 4.7K in parallel with a 680K resistor.
Don't worry about the actual ratio. Just consider it at about
100:1. Again, if we disregarded the 680K effect, we would be
in error (but not very much). Now, take a 1% slice ("Sliver")
off of the 4.7K (which is 0.047K), and we wind up with an approximate
value of 4.65K as a result. [4.67K actual, for 99.6% accuracy]
Example #3: Place a 3.9K in parallel with a 2.7Meg. Now,
we know that the 2.7Meg is not going to affect the actual value
by very much. In fact, it will only drop the 3.9K by about 0.1%,
which is only a drop of about 3.9ohms from the 3.9K. We would
do this by taking off a slice of that 0.1% (3.9ohms) from the
3.9K, to give us an approximate value of 3.89K. [3.89K actual,
for 100% accuracy]
You may have noticed some variations in what
I described, vs what was shown in the schematics, as well as
the fact that if you carried out the calculations with a calculator,
you would find some apparent discrepancies. But remember that
these are "approximations", and some
allowable slop (if you want to call it that ) is not a
problem! PDF File of all 3 Examples (20Kb)
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.... ddf ... 03/11/2007