State that the combined resistance of resistors in parallel is less than that of any single resistor in that circuit

4.3.2 Series and Parallel Circuits

Learning Objective

State that the combined resistance of resistors in parallel is less than the resistance of any single resistor in that circuit.

Key Concepts

• Resistors in series share the same current; the total resistance is the sum of the individual resistances.
• Resistors in parallel share the same voltage; the total (equivalent) resistance is found using the reciprocal rule.

Series Circuits (Brief Review)

For resistors \$R1, R2, \dots , R_n\$ connected end‑to‑end:

\$R{\text{series}} = R1 + R2 + \dots + Rn\$

The current \$I\$ is the same through each resistor, and the total voltage is the sum of the individual voltage drops.

Parallel Circuits

When resistors are connected across the same two points, they are in parallel. Each resistor experiences the same voltage \$V\$, but the currents through them can differ.

Combined (Equivalent) Resistance

The reciprocal of the equivalent resistance \$R_{\text{eq}}\$ is the sum of the reciprocals of the individual resistances:

\$\frac{1}{R{\text{eq}}}= \frac{1}{R1}+ \frac{1}{R2}+ \dots + \frac{1}{Rn}\$

Solving for \$R_{\text{eq}}\$ gives the combined resistance of the parallel network.

Why \$R_{\text{eq}}\$ Is Always Less Than Any Single Resistor

1. Each term \$\frac{1}{R_i}\$ is positive.
2. Adding another positive term to the sum \$\frac{1}{R_{\text{eq}}}\$ makes the total larger.
3. A larger denominator (the sum of reciprocals) means a smaller \$R{\text{eq}}\$ because \$R{\text{eq}} = \frac{1}{\text{sum of reciprocals}}\$.
4. Therefore, adding a resistor in parallel can only decrease the equivalent resistance, never increase it.

Formula Summary

Worked Example

Three resistors \$R1 = 12\ \Omega\$, \$R2 = 18\ \Omega\$, and \$R_3 = 30\ \Omega\$ are connected in parallel. Find the equivalent resistance and comment on its size relative to the individual resistors.

1. Write the reciprocal sum:

   \$\frac{1}{R_{\text{eq}}}= \frac{1}{12} + \frac{1}{18} + \frac{1}{30}\$

2. Find a common denominator (180):

   \$\frac{1}{R_{\text{eq}}}= \frac{15}{180} + \frac{10}{180} + \frac{6}{180}= \frac{31}{180}\$

3. Invert to obtain \$R_{\text{eq}}\$:

   \$R_{\text{eq}} = \frac{180}{31}\ \Omega \approx 5.8\ \Omega\$

4. Observation: \$5.8\ \Omega\$ is smaller than each of \$12\ \Omega\$, \$18\ \Omega\$, and \$30\ \Omega\$, confirming the objective.

Key Take‑away

In a parallel network, the equivalent resistance is always less than the smallest individual resistor because the reciprocal sum always exceeds the reciprocal of any single resistor.