Engineering · Electrical Engineering · Circuit Analysis
Resistors in Series and Parallel Calculator
Calculate the total equivalent resistance of up to five resistors connected in series, parallel, or a mixed series-parallel combination.
Calculator
Formula
For series connections, the total resistance R_series is the direct sum of all individual resistances R_1 through R_n. For parallel connections, the reciprocal of the total resistance R_parallel equals the sum of the reciprocals of each individual resistance. In a mixed network, series and parallel sub-groups are resolved step by step until a single equivalent resistance remains.
Source: Kirchhoff's Circuit Laws and Ohm's Law — IEEE Standard 315, Graphic Symbols for Electrical and Electronics Diagrams.
How it works
When resistors are connected in a circuit, they interact to present a single equivalent resistance to the power source. The way they are connected — series, parallel, or a combination — determines how that equivalent resistance is calculated. Getting this right is critical for ensuring components operate within safe ratings, achieving the desired current distribution, and managing thermal dissipation effectively in any electronic or electrical system.
For series connections, each resistor is placed end-to-end so the same current flows through all of them. The equivalent resistance is simply the arithmetic sum: Rtotal = R1 + R2 + ... + Rn. For parallel connections, resistors share the same two nodes, so the same voltage appears across each one. The reciprocal rule applies: 1/Rtotal = 1/R1 + 1/R2 + ... + 1/Rn. In mixed networks, the circuit is broken into series and parallel sub-groups, each simplified step by step until one equivalent value remains. This calculator supports two common mixed topologies: a series group combined in parallel with a fifth resistor, and a parallel group combined in series with a fifth resistor.
Practical applications of equivalent resistance calculations include selecting current-limiting resistors for LED arrays, designing voltage dividers, analyzing motor winding impedances, sizing protection resistors in sensor circuits, and verifying load conditions in power supply designs. If a supply voltage is entered, the calculator also outputs total current (using Ohm's Law: I = V/R) and total power dissipated (P = V²/R), giving a complete first-order circuit analysis in one step.
Worked example
Example 1 — All in Parallel: Three resistors of 100 Ω, 220 Ω, and 330 Ω are connected in parallel across a 12 V supply.
Step 1: Apply the reciprocal formula.
1/Req = 1/100 + 1/220 + 1/330 = 0.01000 + 0.00455 + 0.00303 = 0.01758 S
Step 2: Invert to get equivalent resistance.
Req = 1 / 0.01758 = 56.87 Ω
Step 3: Calculate total current.
I = V / Req = 12 / 56.87 = 0.211 A (211 mA)
Step 4: Calculate total power.
P = V² / Req = 144 / 56.87 = 2.53 W
Example 2 — All in Series: The same three resistors in series yield Rtotal = 100 + 220 + 330 = 650 Ω. With the same 12 V supply, current = 12/650 = 18.46 mA and power = 144/650 = 0.221 W — demonstrating clearly how series configurations dramatically increase resistance and reduce current compared to parallel arrangements.
Limitations & notes
This calculator assumes ideal resistors with purely resistive impedance and no tolerance variation, temperature drift, frequency dependence, or parasitic inductance or capacitance. Real-world resistors carry manufacturing tolerances (typically ±1% to ±5%) that cause the actual equivalent resistance to deviate from the calculated value. At high frequencies, the inductive and capacitive parasitics of physical resistors become significant and this DC-only model no longer applies. The mixed-topology modes assume a specific network structure (R1–R4 forming the group, R5 as the external element); more complex ladder networks or bridge circuits require dedicated network analysis tools or simulation software such as SPICE. Power dissipation values assume the full supply voltage appears across the equivalent resistance, which is valid only for simple single-source networks.
Frequently asked questions
Why is parallel resistance always less than the smallest individual resistor?
Each additional parallel path provides an alternative route for current, effectively increasing the total cross-sectional area through which current can flow. Mathematically, adding any positive term to the reciprocal sum 1/R_total always increases the sum, which means R_total always decreases. This is directly analogous to adding lanes to a highway reducing overall traffic resistance.
What is the equivalent resistance of two identical resistors in parallel?
For two identical resistors R in parallel, the formula simplifies to R/2. For example, two 100 Ω resistors in parallel give exactly 50 Ω. This is a useful shortcut in circuit design when you need a specific resistance value that is half of a standard available value.
How do I handle more than five resistors in series or parallel?
For series networks, simply add all resistor values together — you can do this in groups and then sum the group totals. For parallel networks, accumulate the reciprocal sum across all resistors and invert at the end. The mathematics scales linearly, so breaking a large network into manageable sub-groups is the standard manual approach.
Does the order of resistors matter in a series or parallel circuit?
No — both the series sum (R1 + R2 + ... + Rn) and the parallel reciprocal sum (1/R1 + 1/R2 + ... + 1/Rn) are commutative and associative operations. The equivalent resistance is identical regardless of the physical order in which resistors appear along the series string or parallel branches.
Can this calculator be used for impedances in AC circuits?
The same series and parallel formulas apply to complex impedances (Z) in AC circuits, but only when all elements are purely resistive (i.e., no capacitors or inductors are present). If reactive elements are involved, you must work with complex phasors and the magnitudes do not simply add — a dedicated AC impedance calculator should be used instead.
Last updated: 2025-01-15 · Formula verified against primary sources.