Introduction to Parallel Resistance Circuits

In this introduction to parallel resistance circuits, we will explain the three key principles you should know:

1. Voltage: The Voltage is Equal Across All Components

In a parallel circuit, all components are connected across the same two points. As a result, the voltage is the same across each component, regardless of its resistance. This uniform voltage is a defining characteristic of parallel circuits.

Analogy: Imagine a river splitting into several branches; each branch experiences the same water level difference.

Figure 1. Parallel circuit with a battery and three resistors.

In the circuit shown in Figure 1, nodes 1, 2, 3, and 4 are electrically identical, as are nodes 5, 6, 7, and 8. Consequently, the voltage across resistors R1, R2, and R3 is identical and equal to the voltage across the battery, which is 1 V.

2. Current: The Total Circuit Current Equals the Sum of Individual Branch Currents

The total current in a parallel circuit is the sum of the currents through each individual branch. Each branch has its own current, determined by its resistance and the shared voltage:

$$I_{\text{total}} = I_1 + I_2 + I_3 + \dots$$

Implication: Branches with lower resistance carry higher currents, while higher resistance branches carry lower currents.

Figure 2. Current distribution in parallel circuit.

In the circuit of Figure 2, we apply Ohm's law to each resistor to find its current because we know the voltage across each resistor is 1 V.

$$I_{R_1} = \frac{V_{R_1}}{R_1} = \frac{1\ \text{V}}{1\ \text{}\Omega} = 1.0\ \text{A}$$

$$I_{R_2} = \frac{V_{R_2}}{R_2} = \frac{1\ \text{V}}{2\ \text{}\Omega} = 0.5\ \text{A}$$

$$I_{R_3} = \frac{V_{R_3}}{R_3} = \frac{1\ \text{V}}{5\ \text{}\Omega} = 0.2\ \text{A}$$

The total current which is current flows from point 7 to point 8 is equal to the sum of the branch currents through R1, R2 and R3.

$$I_{Total} = I_{R_1}+I_{R_2}+I_{R_3}= 1.0 + 0.5 + 0.2 = 1.7 A.$$

3. Resistance: The Total Resistance is Less Than Any Individual Branch Resistance

The total resistance of a parallel circuit is always less than the resistance of the smallest individual branch. This is because the multiple paths for current reduce the overall opposition to flow. The total resistance can be calculated using the formula:

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$$

Concept: Adding more branches decreases total resistance.

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