Parallel DC Circuits

Introduction

A parallel circuit is formed when two or more circuit elements are connected between the same two nodes. In such a configuration, the voltage across each branch of the circuit is the same. Two basic network configurations are commonly used in electrical circuit analysis: series circuits and parallel circuits. Understanding these configurations is essential for analyzing more complex electrical networks. In general, elements are said to be in parallel if they share two common connection points in a circuit.

Meaning of Parallel Connection

Two elements, branches, or circuits are said to be in parallel if they have two points in common. When resistors are connected in parallel, each resistor experiences the same voltage but may carry different currents depending on its resistance value. For example, if resistors $R_1$, $R_2$, and $R_3$ are connected in parallel, the voltage across each resistor is identical.

• Fig. 1(a): the two resistors are in parallel because they are connected at points a and b.
• Fig. 1(b): Resistors $R_1$ and $R_2$ are in parallel because they again have points a and b in common. $R_1$ is not in parallel with $R_3$ because they are connected at only one point (b). Further, $R_1$ and $R_3$ are not in series because a third connection appears at point b. The same can be said for resistors $R_2$ and $R_3$.
• Fig. 1(c): Resistors $R_1$ and $R_2$ are in series because they have only one point in common that is not connected elsewhere in the network. while ($R_1 + R_2$) are in parallel with $R_3$, because their combined effect goes parallel with $R_3$. Resistors $R_1$and $R_3$ are not in parallel because they have only point a in common. In addition, they are not in series because of the third connection to point a.

Furthermore, even though the discussion above was only for resistors, it can be applied to any two-terminal elements such as voltage sources and meters.

Parallel Resistors in a Circuit

For resistors in parallel as shown in [Fig. 2], the total resistance is determined from the following equation: This equation shows that the equivalent resistance of parallel resistors is always smaller than the smallest individual resistance in the network.

Current Relationship in Parallel Circuits

In a parallel circuit, the total current supplied by the source divides among the different branches. If $I_1$, $I_2$, and $I_3$ represent the branch currents, the total current $I$ is Using Ohm's law, the branch currents can be written as Substituting these expressions into the current equation gives Factoring the voltage $E$: But from Ohm's law, Therefore,

Conductance in Parallel Circuits

Since conductance is defined as $$ G = \frac{1}{R} $$ the total conductance of parallel resistors is simply the sum of the individual conductances. This method is often easier when working with multiple resistors connected in parallel.

Important Property of Parallel Circuits

One important property of parallel circuits is that the equivalent resistance is always less than the smallest resistance in the network. For example, if two resistors of $3\Omega$ and $6\Omega$ are connected in parallel, the total resistance will be less than $3\Omega$. This property provides a quick method to check calculations when solving parallel circuit problems.

Special Case: Equal Parallel Resistors

For equal resistors in parallel, the equation for the total resistance becomes significantly easier to apply. For N equal resistors in parallel, This formula is very useful when analyzing circuits containing identical resistors. If you review the examples above, you will find that the total resistance is less than the smallest parallel resistor. That is, in Example 1, 2 Ω is less than 3 Ω or 6 Ω. In general, therefore, This is particularly important when you want a quick estimate of the total resistance of a parallel combination. Simply find the smallest value, and you know that the total resistance will be less than that value. It is also a great check on your calculations. In addition, you will find that

Conclusion

Parallel DC circuits are widely used in electrical and electronic systems because they allow multiple paths for current to flow. Each branch experiences the same voltage while the total current divides among the branches. Understanding the behavior of parallel circuits is essential for analyzing complex electrical networks and designing reliable power distribution systems.