# Parallel Elements

## Introduction

Two or more components are considered to be in parallel if their “leading” and “trailing” terminals are electrically connected. Because of this electrical connection, the components will have the same voltage across them. Thus, sometimes people will simply say that parallel components are components that have the same voltage across them. While this is true, this statement doesn't completely define what parallel means, because two components may be in different parts of a circuit and yet have the same voltage drop across them. Despite having the same voltage drop, if they aren't directly electrically connected, we would not say they are in parallel.

Figure 1 shows a circuit in which all the components are in parallel. In this case a 3.3V voltage source is in parallel with two resistors, $R_1$ and $R_2$. Because all these components are in parallel, one might say $R_1$ is in parallel with the voltage source, $R_1$ is in parallel with $R_2$, $R_2$ is in parallel with the voltage source, or, of course, all these components are in parallel. Keep in mind that the type of source that is present in the circuit is immaterial—all that matters in order for components to be in parallel is that their terminals are electrically connected.

When a schematic has components drawn in parallel, it's usually easy to recognize that they are electrically parallel, too. However, the orientation of components in the schematic has nothing to do with whether or not the components are (electrically) in parallel. For example, in Fig. 2, the resistors $R_1$ and $R_2$ are said to be in parallel and the same voltage appears across them. This circuit is, in fact, identical to the circuit of Fig. 1 despite being drawn slightly differently.

The circuit of Fig. 3 has three resistors in parallel: $R_2$, $R_3$, and $R_4$. The voltage source and $R_1$ are not in parallel with these resistors.

In the circuit shown in Fig. 4, no two resistors are in parallel. However the voltage that appears across $R_1$ is the same as the voltage that appears across the combination of $R_2$ and $R_3$. The resistors $R_2$ and $R_3$ are in series because the same current passes through each of them. Thus, one might say that “$R_1$ is in parallel with the series resistors $R_2$ and $R_3$.” As discussed in the link available at the box to the right, series resistors can be combined to form a single equivalent resistance. Thus $R_1$ can be described as being in parallel with this equivalent resistance.

## Parallel Resistors

When resistors are in parallel, the resistors can be replaced with a single equivalent resistor. Insofar as the rest of the circuit is concerned, there would be no difference between the parallel resistors or the single equivalent resistor. To illustrate how we obtained the equivalent resistance, consider the two parallel resistors, $R_1$ and $R_2$, shown in the circuit on the left of Fig. 5. In Fig. 5, a voltage source supplies the power but the voltage source is not explicitly shown. Instead, we show ground at the bottom and, at the top of the circuit, the voltage $V$ that the source produces. It is understood that there is a path for current to flow from ground, through the source, and back to the top of the circuit.

Let $I_1$ be the current through $R_1$ and $I_2$ be the current through $R_2$. Given that both resistors have the same voltage $V$ across them, we can use Ohm's Law to calculate the currents:

$\begin{array}{l}{I_1} = \frac{V}{{{R_1}}},\\{I_2} = \frac{V}{{{R_2}}}.\end{array}$

Adding these currents, we get the total current into the two resistors, $I_{\mathrm{tot}}$. This current is given by:

${I_{{\rm{tot}}}} = {I_1} + {I_2} = V\left( {\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}} \right).$

Because we know from Ohm's Law that current is given by voltage divided by resistance, we can think of the total current as being given by $I_{\mathrm{tot}} = V / R_{\mathrm{eq}}$ where $R_{\mathrm{eq}}$ is the equivalent resistance. Equating this expression with the equation above, we can write that the equivalent resistance is given by

$\frac{1}{{{R_{{\rm{eq}}}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}.$

Or, rearranging slightly (i.e., taking the inverse of both sides), we can write:

${R_{{\rm{eq}}}} = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}}}.$

This analysis can be extended to any number of parallel resistors. For $N$ parallel resistors, the equivalent resistance is given by

${R_{{\rm{eq}}}} = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \cdots + \frac{1}{{{R_N}}}}}.$

To find the equivalent resistance for two parallel resistors, using a bit of algebra, you can show that the equivalent resistance is given by the product of the resistances over their sum, i.e.,

${R_{{\rm{eq}}}} = \frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}.$

Keep in mind that this product-over-sum formula only works for two resistors. Another thing to keep in mind is that no matter how many parallel resistors there are, the equivalent resistance must always be less than the smallest individual resistance.

## Important Points

• Parallel components' terminals are electrically connected to each other.
• Because their terminals are in contact, components that are in parallel always have the same voltage across them.
• Parallel resistors can be combined to form a single equivalent resistor.
• The equivalent resistance is always less than the smallest individual parallel resistor.