Series and Parallel Circuits

Circuits on the job are either series, parallel, or a combination of both. Understanding how resistance adds up in each configuration is essential for troubleshooting, calculating loads, and passing your journeyman exam.

Series Circuits

In a series circuit, components are connected end-to-end in a single path. Current has only one way to flow.

Series Rules

1. Current is the same through every component: $I_{\text{total}} = I_1 = I_2 = I_3$
2. Voltage divides across each component: $V_{\text{total}} = V_1 + V_2 + V_3$
3. Resistance adds up directly:

$R_{\text{total}} = R_1 + R_2 + R_3 + \ldots$

Example 1: Three Resistive Loads in Series

Three heating elements of 4 $\Omega$, 6 $\Omega$, and 10 $\Omega$ are wired in series on a 240V circuit.

Total resistance:

$R_{\text{total}} = 4 + 6 + 10 = 20 \text{ }\Omega$

Total current:

$I = \frac{V}{R} = \frac{240}{20} = 12 \text{ A}$

Voltage across each element:

$V_1 = 12 \times 4 = 48 \text{ V} \quad V_2 = 12 \times 6 = 72 \text{ V} \quad V_3 = 12 \times 10 = 120 \text{ V}$

Check: $48 + 72 + 120 = 240$ V. Correct — the individual voltages add up to the source voltage.

Parallel Circuits

In a parallel circuit, components are connected across the same two points. Each component has its own path for current.

Parallel Rules

1. Voltage is the same across every component: $V_{\text{total}} = V_1 = V_2 = V_3$
2. Current divides among the branches: $I_{\text{total}} = I_1 + I_2 + I_3$
3. Resistance follows the reciprocal formula:

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

The total resistance in a parallel circuit is always less than the smallest individual resistance.

The Product-Over-Sum Shortcut (Two Resistors Only)

When you have exactly two resistors in parallel, skip the reciprocal formula and use:

$R_{\text{total}} = \frac{R_1 \times R_2}{R_1 + R_2}$

This is faster on the exam and on the job.

Example 2: Two Parallel Branch Circuits

Two 120V branch circuits each feed a different baseboard heater. Heater A has 16 $\Omega$ resistance, Heater B has 24 $\Omega$.

Total resistance (product-over-sum):

$R_{\text{total}} = \frac{16 \times 24}{16 + 24} = \frac{384}{40} = 9.6 \text{ }\Omega$

Total current from the panel:

$I_{\text{total}} = \frac{120}{9.6} = 12.5 \text{ A}$

Current through each heater:

$I_A = \frac{120}{16} = 7.5 \text{ A} \qquad I_B = \frac{120}{24} = 5 \text{ A}$

Check: $7.5 + 5 = 12.5$ A. Correct — the branch currents add up to the total.

Example 3: Three Parallel Loads

A 120V circuit feeds three receptacle loads: 20 $\Omega$, 30 $\Omega$, and 60 $\Omega$ (all in parallel).

Total resistance using the reciprocal formula:

$\frac{1}{R_{\text{total}}} = \frac{1}{20} + \frac{1}{30} + \frac{1}{60} = \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{6}{60} = \frac{1}{10}$

$R_{\text{total}} = 10 \text{ }\Omega$

Total current:

$I = \frac{120}{10} = 12 \text{ A}$

Answer: The combined resistance is 10 ohms and the total current is 12 amps.

Series vs. Parallel Comparison

Quick Reference: Equal Resistors Shortcut

When all resistors in parallel are the same value:

$R_{\text{total}} = \frac{R}{n}$

where $n$ is the number of equal resistors. Three 30 $\Omega$ loads in parallel:

$R_{\text{total}} = \frac{30}{3} = 10 \text{ }\Omega$

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Common Mistakes to Avoid

1. Adding resistors in parallel like series. In parallel, resistance goes down, not up. Two 10 $\Omega$ resistors in parallel equal 5 $\Omega$, not 20 $\Omega$.
2. Using product-over-sum with three or more resistors. The shortcut only works for exactly two resistors. For three or more, use the reciprocal formula.
3. Forgetting to flip the reciprocal. The formula gives you $\frac{1}{R_{\text{total}}}$ — you must take the reciprocal of your result to get $R_{\text{total}}$.
4. Assuming current is equal in all parallel branches. Current divides inversely with resistance — the branch with less resistance carries more current.

Key Takeaways

• In series: resistance adds, current is constant, voltage divides
• In parallel: resistance decreases, voltage is constant, current divides
• The product-over-sum shortcut ($\frac{R_1 \times R_2}{R_1 + R_2}$) saves time for two-resistor parallel combinations
• For $n$ equal parallel resistors: $R_{\text{total}} = R / n$
• Most branch circuits in a building are parallel — each outlet and fixture has the same voltage but draws its own current
• If one load opens in parallel, the others keep working; in series, everything stops

Math for Electricians