So I had a circuit to analyse and I needed to find the equivalent resistor and then I arrived to a confusion. Are R1 and R3 in parallel? Here is the circuit.


simulate this circuit – Schematic created using CircuitLab

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    \$\begingroup\$ It is common to draw the voltage source on the left hand side. If you draw R1 in the middle and V1 and R2 to the left, it becomes more obvious. \$\endgroup\$ – Dampmaskin Oct 15 '19 at 14:50
  • \$\begingroup\$ Yes they are - their ends are connected to each other \$\endgroup\$ – Reversed Engineer Oct 17 '19 at 20:47

Redrawing schematics is a great way to analyze circuits, but also an exercise in why schematics are drawn in particular ways — to more clearly communicate to other engineers.


simulate this circuit – Schematic created using CircuitLab

The rearrangement above should be a little more clear. If you trace a path from one terminal of the battery to the other, you can hopefully see that there are two paths (the split occurs at the junctions on either side of R1).

Series means one-after-another current flow (like a series of events or a television serial). Parallel means that current flows through two or more components at the same time (proportional to the component values).

Just as when you measure voltage, where the value depends on your reference point, components can be series or parallel depending on what you are comparing them to:

You could say:

  • R2 is in series with the voltage source, or
  • R2 forms a series-parallel circuit with R1 and R3, or
  • R1 and R3 are parallel with each other, or
  • R1 and R3 are parallel with a voltage source and some resistance R2

I see a few answers already, but none provides the definition of "parallel":

Two two-pins components are in parallel when the voltage across them is the same

Conversely, with "series", you have:

Two two-pins components are in series when the current through them is the same

Bear in mind that "the same" literally means "the same" in this context. If you have two resistors with two different voltage generators connected to them, and the generators provide the same voltage, then the voltage across the resistors will numerically be the same, but it won't be the same voltage.

The definitions above also solve the confusion if you only have two components, as highlighted in the comments to this answer. In that case, the components are both in series and in parallel, since the voltage across them and the current through them is the same.

Being in series and in parallel with something else at the same time is not impossible, a very common example can be a voltage generator and its load.

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    \$\begingroup\$ While the statements you make about voltage in parallel and current in series being "the same" are true, these are necessary but not sufficient conditions - there are circuits where components which are not in parallel nevertheless have the same voltage drop across them. \$\endgroup\$ – Stobor Oct 16 '19 at 6:32
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    \$\begingroup\$ This seems like an awfully roundabout way of saying things. Why say "A and B are parallel if the difference-in-potential-between-the-two-termini-of-A is the difference-in-potential-between-the-same-two-termini as the difference-in-potential-between-the-two-termini-of-B" when you could just say "A and B are parallel if the two termini of A are the same two termini as the two termini of B"? If the termini matter and the voltage doesn't matter, why mention the voltage at all? \$\endgroup\$ – Tanner Swett Oct 16 '19 at 17:02
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    \$\begingroup\$ This is a confusing way to look at it, especially for a beginner. In a beginner's mind, I could see easy confusion with your definition that would make R2 seem to be parallel. Yes, the voltage across them is the same and yes, it comes from the same separation of charge, but that's just the result of them sharing terminals. How can you differentiate between the voltage drop across two identical resistors in two identical circuits without it being arbitrary in order ensure that they aren't "the same"? The better definition, in which they share terminals, doesn't require anything arbitrary. \$\endgroup\$ – gormadoc Oct 16 '19 at 20:14
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    \$\begingroup\$ Another issue is that it doesn't work for non-parallel loads that don't have any voltage drop. In both cases the voltage is exactly the same, both in magnitude and origin. Saying R1's separation of no charge is different from R2's separation of no charge requires a look at their terminals, necessitating the easier definition again. \$\endgroup\$ – gormadoc Oct 16 '19 at 20:16
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    \$\begingroup\$ Because you delineate between "same" voltage and "equal" voltage in an unnatural manner. To a beginner, they see one voltage source and that it's doing all the work. In my experience with Intro to Circuits students, they don't always see that the voltage across one component is different from another when they are hooked up differently. For a more clear explanation, consider removing R3 and making R1=R2. Now the voltages are the same, exactly V/2. The source is the same. They are still not in parallel. \$\endgroup\$ – gormadoc Oct 17 '19 at 17:12

Yes, because there are two different flows:

  1. V1 -> R2 -> R1 -> V1
  2. V1 -> R2 -> R3 -> V1

R2 and the combination (R1, R3) are in series, but R1 and R3 are in parallel.


R1 and R2 have their ends connected to the same nodes.


simulate this circuit – Schematic created using CircuitLab

Figure 1. Remove R2 and V1 and it becomes very obvious that R1 is in parallel with R3.

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    \$\begingroup\$ Yes but no, if you remove R2 and V1 it becomes ambiguous. It also looks like R1 and R3 are in series. :) \$\endgroup\$ – Dampmaskin Oct 15 '19 at 14:42
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    \$\begingroup\$ How is a drawing with single path suppose to show parallelism, something that requires more than one paths? \$\endgroup\$ – ikegami Oct 16 '19 at 5:32
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    \$\begingroup\$ It's easier to see for some people that, but removing the voltage source from a network makes all definitions about serial and parallel meaningless, so it's likely to add more confusion than clarification. Topological transformations (like moving the middle branch to the left) are much more helpful. \$\endgroup\$ – toolforger Oct 16 '19 at 7:46

Yes, R1 and R3 are in parallel. Both their ends land at the same places. As a mains electrical guy I'm not supposed to presume wires are zero ohms, but if I do, this becomes a fairly simple matter. Conductance = 1/resistance.

R1 (50.5 ohms) has a conductance of 0.01980198 siemens.
R2 (55.83 ohms) has a conductance of 0.01791152 siemens.

In parallel, conductances simply add. So 0.0377135 siemens.
Stated in resistance, R1/3 is 26.52 ohms.

R1/3 and R2 are in series. In series, resistances simply add. 26.52+1.54 = 28.06 ohms.

This shunts a 10V constant-voltage supply, so we can plug 10V into Ohm's Law and done.


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