The best equation is the generalized form. It’s the correct tool to use for any number for parallel resistors. The patterns are ingrained, and we don’t need to stop and consider if the circuit has 2 or 10 parallel components. Either way, the algorithm is the same.
Canonical Article: Current Divider Formula and Derivation
Learning Companion (Q&A): Explore All Questions
The General Form (recommended)
The equation is the best solution as it represents the current divider in its most abstract form.
I_{R_X} = I_{Total} \left( \frac{R_{Total}}{R_X} \right)
Where:
- I_{R_X} is the current through the resistor of interest
- I_{Total} is the total current
- R_{Total} is the equivalent resistance of the parallel resistor combination
- R_X is the resistor of interest
Two parallel resistors equation
This is the common textbook presentation for two resistors.
I_{R_1} = I_{Total} \left( \frac{R_2}{R_1 + R_2} \right)
I_{R_2} = I_{Total} \left( \frac{R_1}{R_1 + R_2} \right)
Where:
- I_{R_1} is the current through resistor R_1
- I_{R_2} is the current through resistor R_2
- I_{Total} is the total current
- R_1 and R_2 are the resistances of the respective branches
As a teacher, the most common error I encountered was placing the wrong resistor in the numerator. This is easy to do as there is a crossing pattern such as when we calculate R1’s current with R2 in the numerator.
Error-prone equation
There is a third version that I strongly recommend you avoid for the reasons listed in the canonical article. For convenience, here is the amplifying text:
Occasionally we encounter a third variation that attempts to generalize the previously mentioned two parallel resistor equation. Personally, I do not like this equation as it requires a picture to explain. It also adds an unnecessary cognitive load, requiring us to differentiate between the “equivalent parallel circuit resistance” and the “equivalent parallel resistance of the circuit excluding the resistor of interest.
Related Safety Note Regarding Parallel Resistors
When I was teaching my students would poke fun at me by saying, “current takes the path of least resistance.” Words can’t explain how much that statement bothers me, as it is fundamentally wrong and can even lead to safety hazards. For the record, current takes all paths, and you don’t want to be in any of those paths.
Tech Tip: There is an old truism implying that we learn the Fall semester material in the spring. That certainly applies to the 1st semester DC voltage divider. The true muscle memory is formed in the 2nd semester after performing hundreds of equations using AC circuit analysis. While DC and AC are related; we must now account for the complex impedance of inductors and capacitor.
Article by Aaron Dahlen, LCDR USCG (Ret.), Application Engineer at DigiKey. Author bio.