Parallel Circuits

Lesson 3 of 4: Electrical Circuits

In this lesson:

• Identify parallel circuits and why branches are independent
• Calculate equivalent resistance:
• Apply Kirchhoff's Current Law to find all branch currents

Learning Objectives for This Lesson

By end of lesson, you will be able to:

1. Identify a parallel circuit and explain branch independence
2. Calculate using
3. Find branch currents and total current
4. Apply KCL to verify results
5. Explain why household circuits use parallel wiring

Series vs. Parallel — Cut One Wire

• Series: cut one wire → all stop
• Parallel: cut one branch → other branches continue

What Makes a Circuit "Parallel"

• Components share the same two nodes — multiple independent paths
• Each branch operates at the full source voltage
• Voltage is the same across every branch:
• Current divides among branches:

Parallel Rules vs. Series Comparison

Quick Check: Independent Branches in Parallel

Two light bulbs are connected in parallel to a 9 V battery.

If one bulb burns out (its branch opens):

• What happens to the other bulb?
• What is the voltage across the functioning bulb?

Equivalent Resistance Formula for Parallel Circuits

• For two resistors: (product over sum shortcut)
• is always less than the smallest individual resistance
• Adding more parallel branches → decreases → more total current

Why Use the Reciprocal Resistance Formula?

Derivation from :

Worked Example: Two Parallel Resistors

Given: , in parallel

Product/sum shortcut:

Verify: ✓ (less than the smallest)

Worked Example: Three Parallel Resistors

Given: , , in parallel

Verify: ✓

Quick Check: Verify R_eq Is Less

and in parallel.

must be less than 6 Ω. What is it?

Use the product/sum shortcut.

Branch Currents:

• Each branch is connected across the same source voltage
• Branch current is determined independently by Ohm's Law
• Smaller → larger branch current (current divider pattern)

Kirchhoff's Current Law at Every Node

• KCL is a statement of charge conservation
• Charge cannot accumulate at a junction — what flows in must flow out
• Use KCL as a verification step after finding all branch currents

Worked Example: Full Parallel Analysis — Step 1

Circuit: , , ,

Step 1: (from the earlier worked example)

Worked Example: Steps 2–3 — Branch Currents

Step 2:

Step 3: · ·

Worked Example: Step 4 — KCL Verification

Current divider pattern:

• Smallest (20 Ω) → largest (0.6 A)
• Largest (60 Ω) → smallest (0.2 A)

Quick Check: Which Branch Carries More?

In a parallel circuit with two branches:

, ,

Which branch carries more current — or ?

Predict first, then calculate.

Guided Practice: Find All Branch Currents

Circuit: , in parallel,

Given: (verify this is less than 10 Ω)

Find: , , . Verify with KCL.

Practice: Full Parallel Circuit Analysis

Solve completely:

, , ,

Find: , , , , . Verify with KCL.

Use all four steps. Verify before proceeding.

Practice Answers with KCL Verification

R_eq: →

I_total:

Branches: , ,

KCL: ✓

Household Circuits: Parallel in Action

Every outlet connects in parallel to the 120 V supply — same voltage at every branch.

Why Parallel — Two Essential Reasons

1. Independence: Each appliance gets full 120 V regardless of others

   • Series wiring would divide the voltage — each device gets a fraction

2. Failure isolation: One failed device doesn't affect others

   • Series wiring: one bad device breaks the entire house

Worked Example: Will the Breaker Trip?

Circuit: 120 V, 20 A breaker

Total: → Breaker trips!

Key Takeaways: Parallel Circuit Essentials

✓ Parallel: multiple paths, same across all branches,

✓ Smaller → larger branch ; (KCL)

✓ always less than smallest branch

Watch Out: Avoid These Three Errors

Parallel branches don't get "weaker" current — each branch gets full source voltage.

is not the average — it's always less than the smallest . Verify every time.

KCL and KVL are universal — both apply to any circuit, not just parallel or series.

What Comes Next: Electric Power

sec-19-4: Electric Power

• How much power does each component dissipate?
• Three forms: , ,
• Why transmission lines use high voltage

Power analysis applies to both series and parallel circuits.