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Parallel Circuits

In a parallel circuit there are multiple paths for current to follow. This is like a water pipe that enters a house then splits off to the kitchen and the bathroom. Once the water is used it enters the drains then all goes down the same sewer pipe. This is also like a parallel circuit because, eventually, all of the paths of a parallel circuit come back together to return to the battery.

A Parallel Circuit

A Parallel Circuit. The current splits up among multiple paths

A parallel circuit is not a parallel circuit because the resistors are drawn side-by-side, but because the circuit splits into multiple paths. A circuit with a single current path is a series circuit. A circuit with multiple current paths is a parallel circuit.

The Same Parallel Circuit

The same parallel circuit as above. Components in a parallel circuit don't need to be drawn parallel to each other. Drawing components end-to-end does not make them a series circuit. None of these resistors are in series with each other. The current splits into multiple paths so it is a parallel circuit.

Notice that in the first circuit the current path splits three ways, through the 10 ohm, 15 ohm and 5 ohm resistors. In the second circuit the current still splits three ways through the same three resistors. The 10 ohm and 15 ohm resistors are not in series with each other just because they are drawn in a straight line. The 10 ohm and 15 ohm resistors are in parallel with each other.

Since there are multiple paths for the current to follow, the total resistance of a parallel circuit is less than any single resistance in the circuit. For example, if you have a 10-inch water pipe, laying another 10-inch pipe next to it will allow twice as much water to flow as the single pipe. This means the resistance of the two pipes together is less than the resistance of either pipe alone. If you have two equal resistors in parallel, together, they will carry twice as much current as either resistor alone. Therefore, two equal value resistors in parallel will have half the resistance of either resistor alone. For example, two 100 ohm resistors in parallel will have a resistance of 50 ohms.

Two 100 ohm resistors in parallel total 50 ohms.

Two 100 ohm resistors in parallel total 50 ohms.

If you have two unequal resistors in parallel, calculating the total resistance is a little more complicated and will be discussed in a moment. However, the total resistance will always be less than the lowest resistance. For example, let's say you have a 10-inch diameter water pipe that can carry 1,000 gallons per hour. Now lay a 5-inch pipe next to it, that pipe can carry 250 gallons per hour.[1]Together the pipes carry 1,250 gallons per hour (gph). They have less resistance together (can carry more water) than either one alone. Notice that the larger pipe carries more water (1,000 gph vs 250 gph) and therefore has less resistance. Together the pipes have even less resistance than the largest pipe. Together they carry 1,250 gph where the largest pipe alone only carries 1,000 gph. Electrical resistors act the same. If you have a 5 ohm resistor in parallel with a 10 ohm resistor, the total resistance will be something less than 5 ohms.

If you have two resistors in parallel, the total resistance can be calculated by multiplying the values, then dividing that product by the sum of the values. This is known as the product-over-sum method. It is expressed by the following formula:

Product Over Sum

For example, if you have 10 ohms and 5 ohms in parallel the total resistance would be calculated as follows:

Product Over Sum Example

Notice that this is just a little less than 5 ohms, the lowest individual resistance.

The product-over-sum method only works for two resistors. If you have three resistors, apply the method to two of the resistors, then apply it again with the third resistor and the outcome of the first two. For example, if you have two 100 ohm resistors and one 50 ohm resistor in parallel:

Three Equal Resistors

The total resistance would be calculated as follows:

First, calculate the total resistance for any two of the three resistors. Here, the two 100 ohm resistors are chosen.

Three Equal Resistors Step 1
Example

The result is 50 ohms. Now, this 50 ohms is calculated in parallel with the remaining 50 ohm resistor.

All Three Resistors


The final result is 25 ohms.

The following formula works for any number of resistors in parallel.

Recipricals

Where:

RT = Total resistance of the parallel circuit
R1, R2 and R3 = each of the resistors in the circuit

Also may be expressed as:

Another Look

If you have a 10 ohm resistor, a 20 ohm resistor and a 30 ohm resistor this would be calculated as follows:

  1. Take the reciprocal of each resistance:

1 10 = 0.1
1 20 = 0.05
1 30 = 0.0333

  1. Add these reciprocals together:

0.1 + 0.05 + 0.0333 = 0.183

  1. Take the reciprocal of this sum:

1 0.183 = 5.46ohms.

Exercise:[2]

Calculate the total resistance of the following sets of parallel resistors
1) 5 ohms, 10 ohms _________
2) 33 ohms, 47 ohms ________
3) 5 ohms, 10 ohms, 15 ohms _________ 4) 33 ohms, 47 ohms, 100 ohms ________
5) 50 ohms, 75 ohms, 80 ohms ________ 6) 200 ohms, 200 ohms, 100 ohms _______

Tip: Remember, any time you have two identical resistors in parallel, the resistance is half of either resistor. For example, two 50 ohm resistors in parallel will have a total resistance of 25 ohms. Two 30 ohm resistors in parallel will have a total resistance of 15 ohms. Apply this to the last problem above. The two 200 ohm resistors in parallel will have a total resistance together of 100 ohms. They together are in parallel with a 100 ohm resistor. This makes 100 ohms in parallel with 100 ohms which has a total resistance of 50 ohms.


Parallel Resistor Formulas - Answers to Questions
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1A pipe with half the diameter of another pipe has the cross sectional area and would carry about the volume of the larger pipe.
21) 3.33 Ω: 2) 19.38 Ω: 3) 2.72 Ω: 4) 16.2 Ω: 5) 21.8 Ω: 6) 50Ω
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