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I wish to understand in the area of parallel circuits why this formula works: $$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdot\cdot\cdot+$$

in particular what is the meaning of $\frac{1}{R_x}$. I understand that it derives from \begin{align} I_T &= I_1 + I_2 + I_3 \\ \frac{V}{R_{t}} &= \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} \\ \frac{V}{R_{t}} &= V \left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\right) \end{align} and so I know how it works. But I'm trying to intuitively understand what is the meaning of the sum of the inverse values of the individual resistors in a parallel circuit. I'm not looking for a mathematical explanation.

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The resistance $R = \frac{U}{I}$.

The inversed resistance $G = \frac{1}{R} = \frac{I}{U}$ is the electrical conductance .

Resistances are additive in serial scenarios, as voltages at the same current are additive.

$$R_\mathrm{T} = \frac{U_\mathrm{T}}{I} = \frac {U_1 + U_2 + U_3}{I} = \frac{U_1}{I} + \frac{U_2}{I} + \frac{U_3}{I} = R_1 + R_2 + R_3$$

Conductances are additive in parallel scenarios, as currents at the same voltage are additive.

$$G_\mathrm{T} = \frac{I_\mathrm{T}}{U} = \frac{I_1 + I_2 + I_3}{U} = \frac{I_1}{U} + \frac{I_2}{U} + \frac{I_3}{U} = G_1 + G_2 + G_3$$

If there are parallel resistors 1 Ohm, 2 Ohm, 5 Ohm, they have the respective conductances 1 S, 0.5 S, 0.2 S, with the summary conductance 1.7 S, which is equivalent to the resistance 1/1.7 Ohm.

For $n$ general resistors:

$$R_\mathrm{T} = \frac 1{G_\mathrm{T}} = \frac {1}{ \sum_{i=1}^{n} G_i } = \frac {1}{ \sum_{i=1}^{n} \frac{1}{R_i} }$$

It is valid even for other passive components like capacitors and inductors, if the respective complex arithmetic and generalized quantities impedance (complex resistance) and admittance (complex conductance) are involved.


Few analogies:

  1. Imagine your car fuel tank has a leaking hole with some flow resistance and gasoline flows out of the tank. Does it helps to slow down leaking if you punch out many other holes ?
  2. If the above tank is leaking at one place, does it help to totally stop leaking if it does not leak on other places at all ?
  3. An insulated wire is a parallel connection of the wire and its insulation. If their resistance were additive, you could not use insulated wires to conduct electricity.
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  • $\begingroup$ And why can't the total resistance just be the sum of the individual resistors? i.e $R_t = R_1 + R_2 + R_3$? $\endgroup$ Mar 16, 2022 at 10:28
  • $\begingroup$ That's the case for multiple resistors in series. In a series connection, the voltages add and the current is the same. In a parallel connection, the voltage is the same and the currents add. $\endgroup$
    – Hilmar
    Mar 16, 2022 at 11:21
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    $\begingroup$ @quadratic sphere Why do you have a fixation of resistance? For series connection you add the inverse of conductances. Do you have a problem with this? For parallel conection you add the conductances. Adding the inverseses of resistances is the same as adding conductances. The answer above explains exacyly "why" is this way. $\endgroup$
    – nasu
    Mar 16, 2022 at 13:00

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