5
$\begingroup$

I find this formula appears often in physics,

To add resistors in parallel, we do $\frac{1}{R_e}=\frac{1}{R_1}+\frac{1}{R_2}+...$

To add springs in series, we do $\frac{1}{k_e}=\frac{1}{k_1}+\frac{1}{k_2}+...$

To add capacitance in series, we do $\frac{1}{C_e}=\frac{1}{C_1}+\frac{1}{C_2}+...$

Is there a name for the method where variables are added to find an equivalence, follow this pattern? Currently I would say "We find equivalent parallel resistance by taking the reciprocal sum of the resistances", but I winged this my self on the spot and I think "reciprocal sum" could be misleading(e.g. $R_e=\frac{1}{R_1}+\frac{1}{R_2}+...$) to someone who doesn't know whats been talked about already. Some similar ideas like root mean square have names, is there a name given to this method?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ Interesting question! As another example, the reduced mass $\mu$ of a two-body system satisfies $\frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2}$. $\endgroup$
    – Michael Seifert
    Commented Apr 30, 2020 at 12:50
  • $\begingroup$ Hello? ........... $\endgroup$
    – Al Brown
    Commented Aug 7, 2021 at 9:28

3 Answers 3

Reset to default
1
$\begingroup$

It is called "parallel addition" or "parallel sum". See Anderson and Duffin's Series and parallel addition of matrices. See also Ellerman's Introduction to Series-Parallel Duality. Others have written about this, too. See Kent E. Erickson, "A New Operation for Analyzing Series-Paralled Networks," IRE Trans. on Circuit Theory, March 1959, pp.124-126.

Improve this answer
$\endgroup$
1
$\begingroup$

“Parallel sum” is the name for that type of adding-up

R_tot = the parallel sum of the R_i’s

R_tot = 1 / ( 1/R_1 + 1/R_2 + ... 1/R_N )

——————

But alternatively, theres always a name for the reciprocal, and those add-up to the equivalent parallel value:

1/R = G, conductance of a resistor. Sum of conductances is total conductance, if parallel

1/k = c, compliance of a spring...

1/C = S, elastance of a capacitor...

Ive always thought these even exist for people who work with them a lot to not have to say “one over” or “parallel sum” etc. Or if considering whether to add something to a design can say “what’s its (eg) elastance?”, knowing immediately how much it will add to the parallel sum

Improve this answer
$\endgroup$
0
$\begingroup$

It is called harmonic mean by mathematicians https://en.wikipedia.org/wiki/Harmonic_mean

Improve this answer
$\endgroup$
7
  • 5
    $\begingroup$ Close but not quite as the factor as there is no factor for the number of items. Harmonic sum would be a nice name but I don't know whether it is ever used. The first example in your link calculates the harmonic mean of $1, 4, 4$ as $2$. Resistors of $1 \Omega, 4 \Omega 4 \Omega$ in parallel would have a resistance of about $0.67 \Omega$. $\endgroup$
    – badjohn
    Commented Apr 30, 2020 at 13:21
  • 1
    $\begingroup$ A reciprocal to the resistance is called conductivity, which is a naturally additive thing. One can invent similar things for the other cases, I guess. $\endgroup$
    – Vladimir Kalitvianski
    Commented Apr 30, 2020 at 14:01
  • 3
    $\begingroup$ @VladimirKalitvianski conductance, not conductivity, is the inverse of resistance, . Conductivity is a material property and the inverse of resistivity. Those pesky suffix endings will get you every time! :) $\endgroup$
    – Bill N
    Commented Apr 30, 2020 at 14:28
  • $\begingroup$ @badjohn Yeah, I thought of harmonic sum, too, but couldn't find a use of it in the cybersphere. $\endgroup$
    – Bill N
    Commented Apr 30, 2020 at 14:29
  • $\begingroup$ @BillN I found one hit for "harmonic sum" but not with the desired meaning. I see that the first sentence of my comment was garbled but you appear to made sense of it. It should have been: Close but not quite as there is no factor for the number of items. $\endgroup$
    – badjohn
    Commented Apr 30, 2020 at 14:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.