Series and Parallel Circuits

Series and parallel are the two fundamental ways to connect components. They behave differently in almost every way — resistance, current, voltage, and failure modes all work differently in each topology. Real circuits combine both; knowing how to recognise and simplify each combination is the core skill of circuit analysis.

Side-by-Side Comparison

PropertySeriesParallel
CurrentSame through every componentDivides — each branch carries its own current
VoltageDivides — each component drops a portionSame across every component
Total resistance (R)R = R1 + R2 + … (always increases)1/R = 1/R1 + 1/R2 + … (always decreases)
Total capacitance (C)1/C = 1/C1 + 1/C2 + … (decreases)C = C1 + C2 + … (increases)
Total inductance (L)L = L1 + L2 + … (increases)1/L = 1/L1 + 1/L2 + … (decreases)
Component failure (open)Breaks entire circuit — all components go offOnly that branch fails — others keep working
Component failure (short)Remaining components share increased voltageCollapses the shared rail — blows supply fuse
Common exampleOld Christmas lights (one fails, all go out)Household wiring (one appliance off, others unaffected)

Series Circuits

Components connected end-to-end so current has only one path through the circuit:

Series Circuit Topology A voltage source connected in series with three resistors (R1, R2, R3) in a single loop. V R1 R2 R3
In a series circuit, the same current flows through all components.

Total resistance:

R_total = R1 + R2 + R3

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

Current: Same everywhere in the loop:

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

Voltage drops: Each resistor drops voltage proportional to its resistance (voltage divider):

V_R1 = I × R1 = 1A × 2Ω = 2V
V_R2 = I × R2 = 1A × 3Ω = 3V
V_R3 = I × R3 = 1A × 5Ω = 5V

KVL check: 2 + 3 + 5 = 10V ✓ (equals supply voltage)

Voltage Divider Formula

Two resistors R1 (top) and R2 (bottom), supply V_in:

V_out = V_in × R2 / (R1 + R2)

Useful for: scaling voltages, setting bias points, generating reference voltages
Limitation: only accurate when load impedance >> R2

Parallel Circuits

Components connected across the same two nodes — current has multiple simultaneous paths:

Parallel Circuit Topology A voltage source connected in parallel with three resistors (R1, R2, R3) providing multiple current paths. V R1 R2 R3
In a parallel circuit, the voltage is the same across all components, while the current divides.

Voltage: Same across all branches (both nodes shared).

Total resistance:

1/R_total = 1/R1 + 1/R2 + 1/R3
(result is always less than the smallest individual resistor)

Two resistors shortcut: R_total = (R1 × R2) / (R1 + R2)

Current per branch:

Example: V=12V, R1=6Ω, R2=4Ω, R3=12Ω

I_R1 = 12/6  = 2A
I_R2 = 12/4  = 3A
I_R3 = 12/12 = 1A

I_total = 2 + 3 + 1 = 6A

Verify: R_total = 1/(1/6 + 1/4 + 1/12) = 1/(2/12 + 3/12 + 1/12) = 12/6 = 2Ω
I = 12/2 = 6A ✓

Current Divider Formula

Two parallel resistors, total current I_total:

I_R1 = I_total × R2 / (R1 + R2)
I_R2 = I_total × R1 / (R1 + R2)

(note: each branch gets the OTHER resistor in the numerator — because higher
resistance means less current in that branch)

Mixed (Series-Parallel) Circuits

Real circuits combine both. The technique is to simplify from the inside out:

Mixed Series-Parallel Circuit Topology A voltage source connected to a series resistor R1, which then splits into parallel resistors R2 and R3. 24V V R1=10Ω R2=20Ω R3=30Ω
A mixed circuit simplifies by resolving the parallel branches first, then treating the whole as a series circuit.
Step 1: Find R2 ∥ R3:
  R_parallel = (20 × 30) / (20 + 30) = 600/50 = 12Ω

Step 2: R1 in series with R_parallel:
  R_total = 10 + 12 = 22Ω

Step 3: Total current from supply:
  I_total = 24V / 22Ω = 1.09A

Step 4: Voltage across parallel section:
  V_parallel = I_total × 12Ω = 13.1V

Step 5: Currents in each parallel branch:
  I_R2 = 13.1V / 20Ω = 0.655A
  I_R3 = 13.1V / 30Ω = 0.436A
  Check: 0.655 + 0.436 ≈ 1.09A ✓

Practical Examples

ApplicationConnectionWhy
LED strings (common in older lights)SeriesOne driver powers many LEDs — but one failure kills the string
LED strips (modern)Parallel sectionsFailure of one LED doesn't kill the rest
Battery banks (increase voltage)SeriesVoltages add: 3 × 3.7V = 11.1V (Li-ion pack)
Battery banks (increase capacity)ParallelCurrent capacity adds, voltage stays same
House wiringParallelEach outlet/light gets full mains voltage independently
Speaker crossover networkMixedHigh-pass (series cap) to tweeter, low-pass (series inductor) to woofer, both in parallel across amp output

References