# Why does the reciprocal of sum of reciprocals relationship appear commonly?

The relationship of 1/x = 1/x1 + 1/x2 + ... seems to appear rather commonly within engineering.

Off the top of my head, I can think of resistors in parallel, capacitors in series, inductors in parallel, and (less relatedly) springs in series as examples of this relationship.

What is this relationship called, if it has a name, and what does it represent?

I have not found any information on this, nor does anything explains why this relationship seems to appear. Every search term I've used either points me towards sums of reciprocals, Ohm's law, or potential dividers.

Resistance isn't the only descriptor of the behaviour of a resistor. That same behaviour could be represented by the reciprocal of resistance, which is conductance (usually called $$\G\$$):

$$G = \frac{1}{R}$$

It's the same information presented in a different form. Instead of describing how well something opposes conduction (resists), the reciprocal describes how well something conducts. Importantly no information is lost in the reciprocation process, and therefore you could employ conductance in any equation, in the place of resistance (with the appropriate changes), and obtain the exact same results.

The total conductance of a group of resistors connected in parallel is the sum of their individual conductances. That's fairly easy to understand intuitively when you picture two resistances in parallel, offering two identical paths for current to take. Their combined ability to conduct is "obviously" twice as much as a single resistor, thus conductance is additive in that scenario:

$$G = G_1 + G_2 + G_3 + \cdots + G_N$$

But since $$\G = \frac{1}{R}\$$, you could replace each of those terms with the equivalent resistance, to get:

$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_N}$$

simulate this circuit – Schematic created using CircuitLab

I'm afraid it's not more esoteric or enlightening than that.

The same goes for capacitance $$\C\$$ with units "coulombs-per-volt" $$\CV^{-1}\$$. Its reciprocal is apparently called "elastance" (I didn't know that, I had to look it up), with the variable name $$\S\$$, and units $$\F^{-1}\$$ or $$\VC^{-1}\$$. Again intuitively, capacitors that are in parallel will store more coulombs of charge for each volt across them (remembering of course that they all have the same voltage across them), implying additivity:

$$C = C_1 + C_2 + C_3 + \cdots + C_N$$

When connected in series, though (and less intuitively than for resistance), it is the reciprocal elastance that accumulates:

\begin{aligned} S &= S_1 + S_2 + S_3 + \cdots + S_N \\ \\ \frac{1}{C} &= \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots + \frac{1}{C_N} \\ \\ \end{aligned}

A more complete answer would require a deep dive into the lumped element model (LEM), consisting of Ohm's law and Kirchhoff's current and voltage laws (KCL and KVL), and an analysis to isolate these behaviours.

In one of the models for mechanical engineering, analogues to current and voltage are force and velocity. Capacitors, inductors and resistors have analogous elements springs, masses and friction (dampers).

The equations relating these mechanical equivalents are very similar to electrical systems. For example, mechanically, for several elements connected end-to-end, intuitively you could see that the rate-of-change of overall length (velocity) is the sum of the rates-of-change-of-length of the individual elements, just as KVL describes the total potential difference to be the sum of individual voltages.

Similarly, in such a series-connected arrangement, the force (tension) in each element is the same, according to Newton's third law, analogous to KCL's stipulation that current be the same in all elements in the path.

Taking capacitance as a further example, and comparing its mechanical analogue, the spring, we'll see more commonalities. The relationship between current through and voltage across a capacitor is:

$$I = C\frac{dV}{dt}$$

The equivalent mechanical model relates a spring's spring constant $$\k\$$, its rate-of-change-of-length (velocity) $$\v\$$ and applied force $$\F\$$ like this:

$$v = \frac{1}{k} \cdot \frac{dF}{dt}$$

The reciprocal of the spring constant is called "compliance", which I'll call (coincidentally) $$\C\$$:

$$C=\frac{1}{k}$$

Due to the equivalence of those two equations, and to the equivalence of KVL when applied to velocity, and KCL applied to force, it's clear that the only difference is variable names. By this argument, I can make the assumption that the combined compliance of a set of springs connected in series will be the sum of their individual compliances:

$$C = C_1 + C_2 + C_3 + \cdots + C_N$$

A group of parallel-connected springs would have a combined spring-constant, and combined compliance:

\begin{aligned} k &= k_1 + k_2 + k_3 + \cdots + k_N \\ \\ \frac{1}{C} &= \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots + \frac{1}{C_N} \\ \\ \end{aligned}

In this way, the LEM can describe macroscopic behaviour of both mechanical and electrical systems, using the same equations and relationships. One would therefore expect to see similarities in the use of reciprocals.

• Not just an analogy, might part of the reason springs and capacitors behave similar be to do with them both being macroscopic effects from electric repulsion of electron clouds. Commented Jul 20 at 14:36

It appears in many contexts, as it's the sum of the harmonic mean of the elements, one of the ways of finding the "most normal value" of a set of values. Typically it's used for rates, such as distance per unit time, and as you say, many electrical situations.

You'll see many examples at the wikipedia article.

It simply means that the unit you're using for this measurement is the reciprocal of the quantity that you're actually adding.

When you put two resistors in series, you're literally adding more resistance, so $$R_{total} = R_1 + R_2$$ When you put two resistors in parallel, you're not adding more resistance. By adding an additional path for current flow, you're adding more conductance, which is the inverse of resistance. Therefore $$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}$$ and a slight rearrangement gives the reciprocal-of-a-sum-of-reciprocals form that you're used to.

This applies to all of the examples that you brought up and more. We have one unit that we conventionally use for each (resistance, impedance, stiffness, etc.) but because those things behave linearly it's equally valid to use inverse units (conductance, susceptance, compliance, etc.). And then, for a "linear combination" of elements like resistors or springs, one of the ways of combining the elements will produce a simple sum in one unit and a "parallel sum" in the other unit, while the other combination reverses the role of the units.

Take 3 resistors in parallel. There are three paths for current to take. Total current is given by Kirchhoff's Current Law. $$I_T = I_1 + I_2 + I_3$$

Apply Ohm's Law to currents. $$\frac {V_T} {R_T} = \frac {V_1} {R_1} + \frac {V_2} {R_2} + \frac {V_3} {R_3}$$

Parallel circuit, so $$\V_T = V_1 = V_2 = V_3\$$. $$\frac {V_T} {R_T} = \frac {V_T} {R_1} + \frac {V_T} {R_2} + \frac {V_T} {R_3}$$

Factor out and divide both sides by $$\V_T\$$, we get: $$\frac {1} {R_T} = \frac {1} {R_1} + \frac {1} {R_2} + \frac {1} {R_3}$$

The recriprocal rule applied to parallel resistors.

It is the result of current getting larger, so total resistance must of decreased, given that resistance must be less than the smallest parallel resistor and greater than 0Ω. Each time a resistor is added in parallel, the total resistance gets smaller, but must be greater than 0Ω (but heading to 0Ω as resistors approach ∞).

This is the mathematical opposite of resistors in series, where resistance gets larger (heading to ∞) and current decreasing. It applies to resistors, inductors or impedances in parallel and capacitors in series.

Series or Parallel circuits typically use opposite rules. If one rule applies to one, it's opposite mathematical rule applies to the other.