Resistors in Series and Parallel
Calculate total resistance for resistors in series (R = R₁+R₂+...) and parallel (1/R = 1/R₁+1/R₂+...).
Essential for circuit design.
The Formulas
Resistors in Series
In a series circuit, resistors are connected end-to-end.
The same current flows through each resistor.
Total resistance is the sum of all individual resistances.
Resistors in Parallel
In a parallel circuit, resistors are connected across the same two points.
Each resistor has the same voltage across it.
Total resistance is always less than the smallest individual resistor.
Variables
| Symbol | Meaning |
|---|---|
| R_total | Total (equivalent) resistance (Ohms, Ω) |
| R₁, R₂, R₃ | Individual resistor values (Ohms, Ω) |
Two-Resistor Parallel Shortcut
This is a convenient shortcut when you have exactly two resistors in parallel.
Example 1
Three resistors of 100 Ω, 220 Ω, and 330 Ω are connected in series. Find the total resistance.
R_total = R₁ + R₂ + R₃
R_total = 100 + 220 + 330
R_total = 650 Ω
Example 2
Two resistors of 60 Ω and 40 Ω are connected in parallel. Find the total resistance.
R_total = (R₁ × R₂) / (R₁ + R₂)
R_total = (60 × 40) / (60 + 40)
R_total = 2,400 / 100
R_total = 24 Ω
When to Use It
Use these formulas when you need to:
- Simplify complex resistor networks into a single equivalent resistance
- Design voltage dividers using series resistors
- Create current dividers using parallel resistors
- Select resistor combinations to achieve a target resistance value
Series circuits are used when you want to divide voltage.
Parallel circuits are used when you want to divide current or reduce total resistance.
Key Notes
- Series: R_total = R₁ + R₂ + … : The same current flows through every series resistor; voltages add. Total resistance is always greater than any individual resistor. One open circuit breaks the entire series chain — Christmas tree light failure mode.
- Parallel: 1/R_total = 1/R₁ + 1/R₂ + …: The same voltage appears across every parallel resistor; currents add. Total resistance is always less than the smallest individual resistor. For two resistors: R_total = R₁R₂/(R₁+R₂) — "product over sum."
- Voltage divider (series): V_out = V_in × R₂/(R₁+R₂): Splits a voltage proportionally. Used in sensor interfaces, biasing circuits, and reference voltage generation. The formula only holds if no significant current is drawn from V_out (loading effect).
- Current divider (parallel): I₁ = I_total × R₂/(R₁+R₂): Note the subscript reversal — R₂ controls I₁ and vice versa. More current flows through the lower resistance path (smaller denominator for that branch).
- Applications: Series-parallel combinations appear in every electronic circuit — resistor networks, attenuator pads, bias networks, Wheatstone bridges (for precision measurement), and the analysis of any multi-resistor circuit using nodal or mesh methods.