Why does the inverse parallel resistor formula work?

neonstrobe said:

Hi

I too use the inverse to illustrate parallel resistors. I'd like to see "s" adopted as a standard inverse r too.

The approach can be illustrated for three or more resistors:

I=(s1+s2+s3)V
=>R = (R1.R2.R3)/(R2.R3 + R3.R1 + R1.R2)

and for (n) in parallel the approach is
top line= product of all
bottom line = sum of the products of (n-1) terms, skipping one resistor in each term each time
You can see above that we skipped R1 in the first term, R2 in the second ...

Click to expand...

Wouldn't you want to show the matrix abstraction of that, rather than stating the product/sum of products? That is the equation that I found a bit confusing when I just started (age 12). It wasn't explained 'why' it works until a college course. Prior to that, it was stated \(\frac{R1\cdot R2}{R1+R2}\) would only work for two resistors.

SullivanTrainingSystems said:

[1] I'm curious to hear what people teach to explain *why* the inverse resistor combination formula works. I understand how parallel resistors affect total resistance and hence current, but I'm interested to learn how people understand why the math formula really works. I have a personal understanding that works for me, but what are any other views?

[2] This is a question of semantics: "Is there any such thing as a parallel circuit, singular?"

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There is a wonderful tool that is almost forgotten by almost everyone except for the power transmission and distribution folks...the reciprocal functions:

Conductance
Susceptance
Admittance.

Conductance= 1/resistance
Susceptance = 1/reactance
Admittance = 1/impedance


Being agile and comfortable with both the "normal" functions and the reciprocals makes working with parallel networks a snap. For parallel circuits you treat the reciprocals just as you would the normal functions for series circuits. Rather than having to memorize the complex parallel impedance formula (which I STILL have to look up!) you siimply take the reciprocal of the values and then plug them into the "normal" impedance formula. The resultant will now be an ADMITTANCE. You can then take the reciprocal again, and return the answer as an IMPEDANCE. Very cool...and more intuitive!

If you get a chance, see my Nov' 98 QST article, "Making the Glass Half Full", where I discuss in detail the elegance of the reciprocal functions.

Hope this helps!

Eric

Most of us have been "brought up" by viewing circuits in terms of how POORLY a conductor works.

Conductance, on the other hand, tells you how WELL a conductor works. If you have multiple CONDUCTANCES (paths for current flow) it's fairly obvious why you would ADD conductances for a parallel circuit.

Eric

I added the equation for three resistors precisely because I too learned that "the formula for two resistors doesn't work for more than two..." at school, before I worked out that this could be extended.

You can play with matrix equations too if you like. Somewhere else there is a thread on the resistor cube. Writing the matrix equations for this makes the job of solving any arbitrary resistor values simple. And you get to play with some means of inverting the matrix such as LUD.

Back to the original question. I too use the water in a pipe analogy. One resistor, one pipe, one flow of water. Another resistor, another pipe = more water. The water shows how well the resistors conduct - which is conductance not resistance, adding the conductances is then simple. The resistor equation just follows from the math.

KL7AJ and neonstrobe,

If resistors were given values in terms of conductance (siemans, mhos) as they could easily be, it would not be so obvious or easy to determine the total conductance or current of a series circuit consisting of several resistors.

Ratch

neonstrobe said:

Hi

I too use the inverse to illustrate parallel resistors. I'd like to see "s" adopted as a standard inverse r too.

The approach can be illustrated for three or more resistors:

I=(s1+s2+s3)V
=>R = (R1.R2.R3)/(R2.R3 + R3.R1 + R1.R2)

and for (n) in parallel the approach is
top line= product of all
bottom line = sum of the products of (n-1) terms, skipping one resistor in each term each time
You can see above that we skipped R1 in the first term, R2 in the second ...

Click to expand...

I'm sure Siemens was a fine gentleman, but I still prefer the MHO. Because when you have Mo' MHO, you have Mo' electricity!

Now you know. I'll say no mho!


Eric

KL7AJ said:

I'm sure Siemens was a fine gentleman, but I still prefer the MHO. Because when you have Mo' MHO, you have Mo' electricity!

Now you know. I'll say no mho!


Eric

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I have a LOT of resistors I need to re-paint the values on now.

someonesdad said:

The math and physics we use are models for experimental knowledge. You could approach the subject from an experimental standpoint: take two resistors and an ohmmeter and measure their separate resistances and combined resistances, both in series and parallel. Perhaps do it with multiple pairs of resistors. These are experimental data and, as such, one may want to look for a pattern in the to see if some rule can explain them.

Now, this may seem too elementary an approach, but remember that all our models need to be based on experimental facts. .

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While this is fundamentally true, one needs to have a healthy respect for WELL-ESTABLISHED law and principle. If I do an experiment that seems to contradict KNOWN physical law, I must first suspect my apparatus or method. I've been crawling around electronics circuits for more than 45 years and I have yet to see anything that violates Ohm's Law.
Experimentation is CRUCIAL, but it has to be based on sound methodology....which is something nobody seems to be teaching any more.

Probably nowhere does one see more "bogus" experimentation than when it comes to antennas. Almost every day, it seems, someone comes out with a "new" antenna design that has more gain than is theoretically possible. In every single instance, these "measurements" have been soundly discredited by reputable antenna engineers, using well-established, starndardized measurement methods.

There ARE indeed several "scientific" discoveries that we can safely dismiss out of hand....such as perpetual motion. Any claim for a perpetual motion machine may be, in all conscience, rubber stamped with a large, bold, bright red B.S. label. It does not even merit experimentation.....it violates known law.

Ranking right up there with perpetual motion is "scalar waves." Poor Nikola Tesla must be spinning in his grave. According to present day tin-foil-hatted Tesla groupies, Tesla discovered some sort of "non-electromagnetic" wave propagation, that propagates unattenuated over vast distances.

This is 100% unmitigated B.S.

ALL of Tesla's writings and inventions can be explained by classical electromagnetic theory. Tesla himself made no claims to the contrary! Sure, Tesla had an understanding of these things well ahead of most of his contemporaries...but he had no arcane knowledge or technology. You can reproduce EVERY Tesla experiment in your back yard.

Ernest Hemingway said, "What every writer needs is a built-in s**t detector...."

The same applies to every engineer.


eric

I have a LOT of resistors I need to re-paint the values on now

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Nah you just read them in a spectral mirror.

Anyone remember what these things were known as .....? - something from long ago I (hopefully correctly) remembered. The word is on the tip of my tongue but wont come out. Hopefully (again) this isn't regarded as a hijack - I think it might be sufficiently related to the OP to pass on that score.

It's potentially an interesting geometrical problem as well.

Was it something like nomograph?

t_n_k said:

Anyone remember what these things were known as .....? - something from long ago I (hopefully correctly) remembered. The word is on the tip of my tongue but wont come out. Hopefully (again) this isn't regarded as a hijack - I think it might be sufficiently related to the OP to pass on that score.

It's potentially an interesting geometrical problem as well.

Click to expand...

Nomograph is correct. I have bunches of nomographs of different types.

By the way, the first time I saw this graphical method was in a slide rule handbook! I don't know WHY it's not published in the ARRL handbook. It's such a slick trick for doing parallel circuits!

Eric

Here's the graphical method for COMPLEX parallel circuits.

Eric

Thanks KL7AJ [Eric],

Yes - The noble slide rule! I owned one once which (to my shame) I think I discarded long ago in favor of an electronic calculator. No problems with the battery running out at a critical moment on 'ye olde slide rule' .... I remember very long ago [at high school actually] solving problems using tables of logarithms and trigonometric values.

Thanks again for the info - much appreciated. The pages have a nicely 'aged' hue.

t_n_k

t_n_k said:

Anyone remember what these things were known as .....? - something from long ago I (hopefully correctly) remembered. The word is on the tip of my tongue but wont come out. Hopefully (again) this isn't regarded as a hijack - I think it might be sufficiently related to the OP to pass on that score.

It's potentially an interesting geometrical problem as well.

Click to expand...

This is an interesting way to find the equivalent resistance of 2 resistors in parallel. I proved that it works for any separation between O1 and O2 (other than 0, of course), can you?

zgozvrm said:

This is an interesting way to find the equivalent resistance of 2 resistors in parallel. I proved that it works for any separation between O1 and O2 (other than 0, of course), can you?

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I think so.

studiot said:

Nah you just read them in a spectral mirror.

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Now that's funny! Didn't know you had in you lad.

t_n_k said:

I think so.

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Nice job! Your way is much more elegant than mine!