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Instead of a single resistor, electric circuits often employ multiple resistors. Let us consider the case where we have two resistors that are denoted as R _{1} and R _{2} .
The two resistors can be connected in an end-to-end manner as shown in Figure 2 (a).
Resistors connected in this manner are said to be connected in series . We may replace a series connection of two resistors by a single equivalent resistance, R _{eq} . From an electrical standpoint, the single equivalent resistance will behave exactly the same as the combination of the two resistors connected in series. The equivalent resistance of two resistors connected in series can be calculated by summing the resistance value of each of the two resistors.
Let us now consider the case where the two resistors are placed side-by-side and then connected at both ends. This situation is depicted in Figure 2 (b) and is called a parallel combination of resistors. Whenever resistors are connected in this manner, they are said to be connected in parallel. The equivalent resistance of two resistors connected in parallel obeys the following relationship.
Once this quantity is calculated, one may easily take the reciprocal of the result to obtain the value of R _{eq} .
The rule governing the determination of the equivalent resistance for the series connection of more than two resistors can be expanded to accommodate any number ( n ) of resistors. For n resistors connected in series, the equivalent resistance is equal to the sum of the n resistance values.
In addition, the rule governing the determination of the equivalent resistance for the parallel connection of more than two resistors can be expanded. For n resistors connected in parallel,
It is clear to see that one’s ability to determine the equivalent resistance of parallel resistors depends upon one’s ability to add fractions. The following exercises are included to reinforce this idea.
Example 1: Two resistors are connected in series. The value of resistance for the first resistor is 5 Ω, while that of the second is 9 Ω. Find the equivalent resistance of the series combination.
Example 2: Consider the two resistors presented in Example 1. Let the two resistors now be connected in parallel. Find the equivalent resistance of the parallel combination.
The lowest common denominator is 45 Ω. So we incorporate it into our analysis.
Example 3: Three resistors of values 2 kΩ, 3 kΩ, and 5 kΩ are connected in parallel. Find the equivalent resistance.
The lowest common denominator for the fractional terms is
We rewrite the original equation to reflect the lowest common denominator
${R}_{\text{eq}}=\frac{\text{30},\text{000}\Omega}{\text{31}}=\text{968}\Omega $
This module has presented how to add fractions using the lowest common denominator method. Also presented is the relationship among voltage, current and resistance that is known as Ohm’s Law. Examples illustrating the use of the lowest common denominator method to solve for equivalent resistances of parallel combinations of resistors are also provided.
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