# Is there a name for the way parallel resistors (and others) are added?

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?

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

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.

“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

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

• 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 $0.67 \Omega$. Commented Apr 30, 2020 at 13:21
• 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. Commented Apr 30, 2020 at 14:01
• @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! :) Commented Apr 30, 2020 at 14:28
• @badjohn Yeah, I thought of harmonic sum, too, but couldn't find a use of it in the cybersphere. Commented Apr 30, 2020 at 14:29
• @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. Commented Apr 30, 2020 at 14:53