Resistances in Parallel

When resistances are placed in parallel, they behave differently than they do in series. One way to look at resistances in parallel is to consider them as conductances instead. In parallel, conductances add up directly, just as resistances add up in series. If you change all the ohmic values to siemens, you can add these figures up and convert the final answer back to ohms.
The symbol for conductance is G. This figure, in siemens, is related to the resistance R, in ohms, by these formulas :

G = 1/R
R = 1/G

Problem 1

Consider five resistors in parallel. Call them R1 through R5, and call the total resistance R as shown in following figure. Let the resistance values be as follows: R1 = 100 Ω, R2 = 200 Ω, R3 = 300 Ω, R4 = 400 Ω, and R5 = 500 Ω. What is the total resistance, R, of this parallel combination?


Five resistors of values R1 through R5, connected in parallel, produce a net resistance R. Illustration for above Problems

Converting the resistances to conductance values, you get: G1 = 1/100 = 0.01 S, G2 = 1/200 = 0.005 S, G3 = 1/300 = 0.00333 S, G4 = 1/400 = 0.0025 S, and G5 = 1/500 = 0.002 S.
Adding these gives G = 0.01 + 0.005 + 0.00333 + 0.0025 + 0.002 = 0.0228 S. The total resistance is therefore R = 1/G = 1/0.0228 = 43.8 Ω.

Problem 2

Suppose you have five resistors, called R1 through R5, connected in parallel as shown in Problem 1 figure. Suppose all the resistances, R1 through R5, are 4.70 kΩ. What is the total resistance, R, of this combination?

When you have two or more resistors connected in parallel and their resistances are all the same, the total resistance is equal to the resistance of any one component divided by the number of components. In this example, convert the resistance of any single resistor to 4700 Ω, and then divide this by 5. Thus, you can see that the total resistance is 4700/5 = 940 Ω.

In a situation like this, where you have a bunch of resistors connected together to operate as a single unit, the total resistance is sometimes called the net resistance. Take note, too, that R is not italicized when it means resistor, but R is italicized when it means resistance!