Series and Parallel Circuits

Series and parallel are the two ways to connect components, and they behave very differently. Understanding how resistance, voltage, and current work in each configuration is the foundation of circuit analysis — everything else builds on these two cases.

Series Circuits

Components connected end-to-end so that current has only one path through the circuit. The same current flows through every component.

    +──[R1]──[R2]──[R3]──+
    |                    |
   [V]                   |
    |                    |
    +────────────────────+

Total resistance:

R_total = R1 + R2 + R3

Example: R1=2Ω, R2=3Ω, R3=5Ω
R_total = 2 + 3 + 5 = 10Ω

Current: Same everywhere in the loop. If V=10V:

I = V / R_total = 10V / 10Ω = 1A
(1A flows through R1, R2, and R3)

Voltage drops: Each resistor drops voltage proportional to its resistance (Ohm's Law). The drops add up to the supply voltage:

V_R1 = I × R1 = 1A × 2Ω = 2V
V_R2 = I × R2 = 1A × 3Ω = 3V
V_R3 = I × R3 = 1A × 5Ω = 5V
Total: 2 + 3 + 5 = 10V ✓

This is a voltage divider — the supply voltage is divided among the resistors. In series circuits, one component failing (open circuit) breaks the entire path.

Parallel Circuits

Components connected across the same two points, giving current multiple paths to flow through simultaneously.

    +────+────+────+
    |    |    |    |
   [V]  [R1] [R2] [R3]
    |    |    |    |
    +────+────+────+

Voltage: Same across all branches — every component sees the full supply voltage.

Total resistance:

1/R_total = 1/R1 + 1/R2 + 1/R3

Example: R1=2Ω, R2=4Ω, R3=4Ω
1/R_total = 1/2 + 1/4 + 1/4 = 1
R_total = 1Ω

(Total is always less than the smallest individual resistor)

For just two resistors in parallel, the product-over-sum shortcut:

R_total = (R1 × R2) / (R1 + R2)

Current: Each branch carries its own current independently. Total current from the supply is the sum:

If V=12V, R1=6Ω, R2=12Ω:
I_R1 = 12/6 = 2A
I_R2 = 12/12 = 1A
I_total = 3A

Check: R_total = (6×12)/(6+12) = 72/18 = 4Ω
I = 12/4 = 3A ✓

In parallel circuits, one component failing (open) doesn't affect the other branches — the rest keep working. This is why household wiring uses parallel circuits.

Mixed Circuits

Real circuits combine series and parallel. Solve by simplifying step by step — find equivalent resistances for parallel groups, then add them in series:

+──[R1]──+──[R2]──+
         |        |
        [R3]     [R4]
         |        |
+────────+────────+

Step 1: R2 ∥ R3 (R2 and R3 are in parallel)
Step 2: Result in series with R1, then parallel with R4