New Method for Parallel Resistor Calculation

Thread Starter

paulfr

Joined Jul 2, 2011
2
I teach HS Physics part of which is Electric Circuits,
I have discovered a rule / technique for parallel resistors that I never encountered
in all my 30+ years in electronics engineering, nor in any textbook on Circuits.
It is what I call " The N + 1 Rule "

We all know the Reciprocal Rule
1 / RT = 1/R1 + 1/R2 + 1/R3 ..... + 1/Rn

AND
we know that for 2 resistors, this becomes the Product over the Sum of the 2 R's

BUT
The N+1 Rule is this
1/ Find N = the ratio of the two R's
2/ Add 1 to it to get N + 1
3/ Divide the largest R by N+1

E.g.
4 and 20 ohms
N = 20/4 = 5
N+1 = 6
RT = Rtotal = 20/6 or 10/3
Check
Product = 80
Sum= 24
RT = 80/24 = 10/3

It is quite useful when the numbers are large and thus the Product is very large.
No need to remember it and then do long division.
e.g.
300 and 50 becomes much easier and thus faster with N+1 than with Product-Sum.
300/50 = 6 ==> Rtotal = 300/7
Check with Product Sum Rule
300 (50) / 300 + 50 = 15000/350 = 300/7

It works even when N is not an integer.
e.g.
500 and 300 ohms
500/300 + 1 = 5/3 + 1 = 8/3
Rtotal = 500 / (8/3) = 1500/8
Check with Product Sum Rule
500(300) / (500 + 300) = 150000 / 800 = 1500/8

Have any of you ever seen this ???

Just curious and wondering why it is not in all the textbooks on Circuits.

Comments solicited
 

bogosort

Joined Sep 24, 2011
571
The N+1 Rule is this
1/ Find N = the ratio of the two R's
2/ Add 1 to it to get N + 1
3/ Divide the largest R by N+1
What you've found is a reformulation of the standard product-over-sum rule:

\(\frac{R_1 R_2}{R_1 + R_2} = \frac{R_1}{\frac{R_1 + R_2}{R_2}} = \frac{R_1}{\frac{R_1}{R_2} + 1}\)

Just curious and wondering why it is not in all the textbooks on Circuits.
Probably because it's easier to remember "product over sum" than the N + 1 reformulation. There's also the fact that the product-over-sums method is "commutative" in R, whereas the N + 1 method is sensitive to R1 >= R2, which introduces another possible error source.

You could try running an informal experiment with your students: teach both methods and see which they use on tests and which has the better success rate.
 

MrAl

Joined Jun 17, 2014
7,888
I teach HS Physics part of which is Electric Circuits,
I have discovered a rule / technique for parallel resistors that I never encountered
in all my 30+ years in electronics engineering, nor in any textbook on Circuits.
It is what I call " The N + 1 Rule "

We all know the Reciprocal Rule
1 / RT = 1/R1 + 1/R2 + 1/R3 ..... + 1/Rn

AND
we know that for 2 resistors, this becomes the Product over the Sum of the 2 R's

BUT
The N+1 Rule is this
1/ Find N = the ratio of the two R's
2/ Add 1 to it to get N + 1
3/ Divide the largest R by N+1

E.g.
4 and 20 ohms
N = 20/4 = 5
N+1 = 6
RT = Rtotal = 20/6 or 10/3
Check
Product = 80
Sum= 24
RT = 80/24 = 10/3

It is quite useful when the numbers are large and thus the Product is very large.
No need to remember it and then do long division.
e.g.
300 and 50 becomes much easier and thus faster with N+1 than with Product-Sum.
300/50 = 6 ==> Rtotal = 300/7
Check with Product Sum Rule
300 (50) / 300 + 50 = 15000/350 = 300/7

It works even when N is not an integer.
e.g.
500 and 300 ohms
500/300 + 1 = 5/3 + 1 = 8/3
Rtotal = 500 / (8/3) = 1500/8
Check with Product Sum Rule
500(300) / (500 + 300) = 150000 / 800 = 1500/8

Have any of you ever seen this ???

Just curious and wondering why it is not in all the textbooks on Circuits.

Comments solicited

Hello there,

Yes i believe that is derived from one of the other formulas for parallel resistors, but there is a truly different way that i came up with several years back, but is also limited as yours is to being able to calculate a fraction in the end. I'll give the same example.

20 Ohms and 4 Ohms:
We know right off that the 4 Ohm resistor is the same as five 20 Ohm resistors in parallel. So we have five 20 Ohm resistors in parallel with one more 20 Ohm resistor which means six 20 Ohm resistors in parallel which means the total resistance is 20/6 or 10/3 which is easy to imagine.

20 and 2 would go the same way, the 2 Ohm is equivalent to ten 20 Ohm units in parallel so we have a total of 11 units in parallel which is 20/11.

So we just convert the lower resistance into as many of the higher resistances we need then realize that we have one more in parallel with that.

None of these methods are that nice though when we have more unusual values.
For a simple example, 9 ohms and 3 ohms, and since 3 Ohms is three 9 ohm'ers we have four 9 ohm'ers in parallel for 9/4 Ohms total.
But it gets a little more unusual when we have say 8 Ohms and 3 Ohms. Then it is hard to convert that 3 Ohms into parallel 8 Ohm resistors, so maybe we can approximate that. And it is the same with the other method:
8/3+3/3=11/3
8/11/3=24/11
Now we have to approximate that 24/11.
We might say that 2*11 is 22, and 0.2 *11 is 2.2, and 22+2.2=24.2 which is close, so the total resistance is approximately 2.2 but you see it still get a little tricky.
 

MrAl

Joined Jun 17, 2014
7,888
I can tell you from experience with my students that half the time they forget to do the reciprocal at the end.
Hi,

Oh ha ha, sorry to hear that but at the same time i cant help but laugh just a little.
Kids: cant live with 'em, cant live with 'em :)

Teach the other way too with product over sum. Then they can choose.

There is one more rendition i should mention for multiple resistors. It's the product of all resistors divided by the sums of products of all resistors taken N-1 at a time.
The three resistor version looks like this:
RT=(R1*R2*R3)/(R2*R3+R1*R3+R1*R2)

and the four resistor version looks like this:
RT=(R1*R2*R3*R4)/(R2*R3*R4+R1*R3*R4+R1*R2*R4+R1*R2*R3)

and so of course the five resistor version looks like this:
RT=(R1*R2*R3*R4*R5)/(R2*R3*R4*R5+R1*R3*R4*R5+R1*R2*R4*R5+R1*R2*R3*R5+R1*R2*R3*R4)

We see we get N products in the numerator and N sums in the denominator.
The N sums in the denominator are made up of all products of N-1 resistors.
 

Hymie

Joined Mar 30, 2018
924
The answer should tell them they made a mistake. :rolleyes:
As someone who went through high school without the advantage of using electronic calculators – I noticed that those following on (who used calculators at school) have a propensity to believe the calculator displayed result without question.

One example I can cite is a young employee who was given the task of finding the mean of multiple sets of 4 numbers. All the numbers ranged from 60 to 65 and were given to 3 decimal places. Using a calculator, the employee had correctly calculated mean of all the sets of figures, except one – the one incorrect mean value for the 4 numbers was given as circa 250. Although once it was pointed out to the employee that the answer was wrong, he immediately knew why.

Without having the drudgery of basic maths (+,-,x, /), it seems strange to me that the saved teaching time was not spent on simple checks that the final calculator displayed result is within the expected range.
 

Ylli

Joined Nov 13, 2015
999
As long as the resistors have a integer ratio, this is the way I do it in my head all the time. Never considered it anything special.

You can do it for non-integer ratio too: for example 50 ohms || 125 ohms....
50 ohms is the same as 5-250 ohm resistors in parallel
125 ohms is the same as 2-250 ohm resistors in parallel
So you have 7-250 ohm resistors in parallel, 250/7 = 35.7 ohms
 
So, you need to spend a min of $25.00 at a store.

Some items you estimate, some you can't count. but I do a two column method. Column 31 is the item price. yep, there are more columns because I'm showing work.

$2.99 3 -1
$5.39 8 39 --->38
$10.50 18 ---->88

So, I've spent $18.88, but I only deal with whole numbers. $ and pennies. I can't "check my work", I don;t need a calculator either.

I agree you definately have to do order of magnitude checks. A yard of concrete is a cubic yard of concrete and sometimes 3/4 is 7/8".
 

atferrari

Joined Jan 6, 2004
4,172
I believe this article brings up the easiest way to calculate an equivalent resistance. Did you really go through the whole thing? In this article, the author has covered almost everything.
@faizan81

I went through the whole article. Now my questions:

a) Could you point at the easiest way, according to you?
b) What procedure have you been using until reading that article?
c) What procedure do you use after reading it?

Interested.
 
Last edited:

BillB3857

Joined Feb 28, 2009
2,541
In my earlier life, part of my job was to conduct technical interviews for potential employment. One question I asked was, "What is the equivalent resistance of a 100 ohm resistor in parallel with a 50 ohm resistor?" I would then watch to see how the applicant attempted to solve the problem. Their method demonstrated two things. Those that used the fact that the low value was the same as two of the higher value in parallel, so the answer would be the higher value divided by 3 displayed two things. One, their understanding, and two, their ability to feel free at using the easiest way to solve a problem rather than the usual text book method.
 

WBahn

Joined Mar 31, 2012
26,398
I can tell you from experience with my students that half the time they forget to do the reciprocal at the end.
The teach them to track their units and take off massive penalties if they fail to do so.

By the midpoint in the semester someone that doesn't track their units through their work gets zero credit for it.

What I found interesting was that the students would at first complain loudly about such a strict units policy (which would initally be up to a 25% penalty for failure to track units) because they would be losing points right and left. But by the midterm exam they would be breathing a sigh of relief because, as they put it, I finally stopped being so picky about units. In reality, the penalties had already tripled by then, but no one was making units mistakes any more, plus they were catching most of their silly algebra mistakes because of how they usually mess up the units.
 

SamR

Joined Mar 19, 2019
2,903
I would draw 2 thermometers on the board. Draw a line across the top of both as the boiling point of water. Another line across the bottom of both as the freezing point. Mark one thermometer as F and the other as C. Mark the top line 212 and 100 and the bottom 32 and 0. and ask them how many each degree of C was equal to in F? And they would sit there like turnips. They could not comprehend there were 180 degrees between freezing and boiling in F much less divide it by 100 in their head. Yeah I taught High School science after being RIFed.
 

MrAl

Joined Jun 17, 2014
7,888
@faizan81

I went through the whole article. Now my questions:

a) Could you point at the easiest way, according to you?
b) What procedure have you been using until reading that article.
c) What procedure do you use after reading it?

Interested.
Hi,

This looks like just a push for another website to me. The only other post by said user at time of this writing reads the same it links to same site with information that is actually less informative than the posts before it.
But i wont judge anyone too harshly just yet giving them time to contribute here first. It could be they just like that site :)
 

atferrari

Joined Jan 6, 2004
4,172
In my earlier life, part of my job was to conduct technical interviews for potential employment. One question I asked was, "What is the equivalent resistance of a 100 ohm resistor in parallel with a 50 ohm resistor?" I would then watch to see how the applicant attempted to solve the problem. Their method demonstrated two things. Those that used the fact that the low value was the same as two of the higher value in parallel, so the answer would be the higher value divided by 3 displayed two things. One, their understanding, and two, their ability to feel free at using the easiest way to solve a problem rather than the usual text book method.
That is just a particular case. Otherwise, you should use the 1/Req = etc... Nothing too relevant, I think.
 

atferrari

Joined Jan 6, 2004
4,172
The teach them to track their units and take off massive penalties if they fail to do so.

By the midpoint in the semester someone that doesn't track their units through their work gets zero credit for it.

What I found interesting was that the students would at first complain loudly about such a strict units policy (which would initally be up to a 25% penalty for failure to track units) because they would be losing points right and left. But by the midterm exam they would be breathing a sigh of relief because, as they put it, I finally stopped being so picky about units. In reality, the penalties had already tripled by then, but no one was making units mistakes any more, plus they were catching most of their silly algebra mistakes because of how they usually mess up the units.
Our professor of thermodynamics at the naval school threatened us with arrest in a dungeon (literally) if we did not track the units along the calculations. Good habit!!

An additional requirement was to work always with pencil (coincident with the work on the bridge on the navigation chart). Regarding this last, for more than 28 years now, I still maintain that habit, throughout my almost 50 notebooks used in my daily work).

20190512_125150 (Personalizado).jpg
 
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