Series and Parallel Resistance Calculator
Calculate equivalent resistance for all-series, all-parallel or reducible mixed resistor networks. Enter independent resistance units, add an optional source voltage and review exact reduction, current, voltage and power results.
- Up to 40 resistors
- Nested S and P groups
- Exact unit conversion
- Zero-ohm checks
Equivalent Resistance Calculator
Choose a network type, enter the resistor values and select each unit. For a mixed network, describe nested series and parallel groups with the guided expression format.
Equivalent resistance across the two network terminals.
Equivalent Resistance Units
| Unit | Equivalent value |
|---|---|
| ohms (Ω) | 650 Ω |
Resistor Results
| Part | Resistance | Voltage | Current | Power |
|---|---|---|---|---|
| R1 | 100 Ω | 1.84615 V | 18.4615 mA | 34.0828 mW |
Calculation Steps
- Add the three series resistances: 100 + 220 + 330 = 650 Ω.
The series result is at least as large as every individual resistor. Values model ideal passive resistors and exclude tolerance, temperature and wiring effects.
How to Use This Series and Parallel Resistance Calculator
- Choose the network type. Select an all-series list, an all-parallel list or a nested mixed series-parallel expression.
- Enter every resistance. Add or remove rows for a simple network. Each row accepts its own unit from picoohms through teraohms.
- Describe a mixed network carefully. Use S(...) for a series group and P(...) for a parallel group. Keep every connection explicit.
- Add source voltage if needed. Leave this field blank for equivalent resistance only. Enter a nonnegative source magnitude to calculate ideal current, voltage and power distribution.
- Choose display units and precision. These settings change the presentation, not the exact internal arithmetic.
- Calculate and review. Check the reduction steps, result bounds, zero-ohm warnings and model limits before using the value.
What Is Equivalent Resistance?
Equivalent resistance is the single resistance that gives the same terminal voltage-to-current relationship as the specified resistor network. Replacing the complete network with that one value preserves its ideal behavior as seen from the two chosen terminals. It does not preserve every internal branch voltage or current.
The calculation depends on connectivity. The same three resistor values produce different answers when they are placed in series, parallel or a nested combination. This tool models ideal, passive and linear two-terminal resistors. It assumes the connections described by the selected mode are exact.
Series and Parallel Resistance Formulas
For resistors in series, the same current passes through each resistor. Their voltage drops add, so their resistances add directly.
For positive resistors in parallel, each branch has the same voltage. Branch currents add, so conductances, the reciprocals of resistance, add.
| Property | Series network | Parallel network |
|---|---|---|
| Shared quantity | Current is the same | Voltage is the same |
| Equivalent formula | Sum the resistances | Invert the reciprocal sum |
| Result bound | At least the largest resistor | Below the smallest positive resistor when at least two branches are present |
| Zero-ohm term | Adds zero to the total | Shorts the ideal parallel group to zero |
| Equal resistors R | nR | R/n |
For two positive parallel resistors, the reciprocal formula simplifies to Req = R1R2/(R1 + R2). The calculator uses the general reciprocal form so the same engine works for any supported branch count.
How to Calculate a Mixed Series-Parallel Network
A reducible mixed network contains identifiable inner groups that are entirely series or entirely parallel. Replace one inner group with its equivalent resistance, then repeat outward until one value remains.
The expression mode makes those groups explicit. S(100, P(220, 330)) means a 100-ohm resistor in series with a parallel group containing 220 ohms and 330 ohms. The inner parallel result is 132 ohms, so the complete equivalent resistance is 232 ohms.
Groups may be nested up to eight levels, with up to 40 resistor leaves. Values without a suffix are read as ohms. You may append pΩ, nΩ, uΩ, µΩ, mΩ, kΩ, MΩ, GΩ or TΩ. ASCII forms such as kohm and Mohm also work.
This grammar does not represent a Wheatstone bridge, delta network or arbitrary mesh. Those circuits may have no sequence of simple series and parallel reductions. They need Kirchhoff equations, nodal analysis, mesh analysis or a suitable network transformation.
Worked Resistance Examples
Example 1: Three resistors in series
Place 100 Ω, 220 Ω and 330 Ω in series. Add the values directly:
With 12 V across the network, the ideal current is 12/650 = 0.0184615 A, or 18.4615 mA. The voltage drops are proportional to the three resistances and add to 12 V.
Example 2: Two resistors in parallel
For 100 Ω and 220 Ω in parallel, use the two-resistor shortcut:
The answer is lower than 100 Ω, the smaller branch resistance. With 10 V applied, the branch currents are 0.1 A and about 0.0454545 A, which sum to the total current.
Example 3: Equal parallel branches
Four equal 1 kΩ resistors in parallel give 1 kΩ/4 = 250 Ω. Each branch carries one quarter of the total current because every branch has the same voltage and resistance.
Example 4: Nested series and parallel groups
For S(2, P(6, 3), 4), the inner parallel group equals 2 Ω. The outer series sum is 2 + 2 + 4 = 8 Ω. At 16 V, total current is 2 A. The parallel group has a 4 V drop, so its branch currents are 2/3 A and 4/3 A.
Voltage, Current and Power Distribution
Source voltage is optional because resistance alone does not determine a circuit current or power. When a finite positive equivalent resistance and source voltage are available, the calculator applies Itotal = V/Req and Ptotal = VI.
Inside a series group, every child receives the same current and its voltage drop is Vi = IRi. Inside a parallel group, every branch receives the same voltage and its current is Ii = V/Ri. Leaf power is calculated with Pi = ViIi.
For a positive network, the calculated values follow the same conservation relationships: series voltage drops sum to the parent voltage, parallel branch currents sum to the parent current and resistor powers sum to total ideal power. Displayed values may show small rounding differences, while internal rational values remain exact.
Zero-Ohm Values and Ideal Short Circuits
A zero-ohm resistor in series contributes zero to the sum. A zero-ohm branch in parallel makes the ideal equivalent resistance of that parallel group zero because current has a path with no modeled resistance.
Equivalent resistance can still be reported exactly. Source analysis needs more care. A nonzero ideal voltage across zero equivalent resistance has no finite current in the ideal model. Zero volts across zero resistance gives an indeterminate 0/0 current. The calculator withholds current and power in both cases instead of displaying Infinity, NaN or an invented short-circuit current.
Real short-circuit current is limited by source impedance, conductor resistance, protection devices, contacts and other nonideal effects. Add known source or wire resistance as an explicit series resistor if it belongs in the intended model.
Inputs represent finite resistance values. The expression parser does not accept OPEN or infinity. Do not replace an open circuit with an arbitrary large number; an open series path is a separate circuit state outside this calculator's finite-resistance model.
Resistance Units and Precision
The SI unit of electrical resistance is the ohm, symbol Ω. One ohm equals one volt per ampere. SI prefixes scale the unit by exact powers of ten, so every input is converted to ohms before the network formula is applied.
| Unit | Symbol | Ohm factor | Example |
|---|---|---|---|
| picoohm | pΩ | 10-12 | 1 pΩ = 0.000000000001 Ω |
| nanoohm | nΩ | 10-9 | 1 nΩ = 0.000000001 Ω |
| microohm | µΩ | 10-6 | 1 µΩ = 0.000001 Ω |
| milliohm | mΩ | 10-3 | 1 mΩ = 0.001 Ω |
| ohm | Ω | 1 | 1 Ω = 1 Ω |
| kiloohm | kΩ | 103 | 1 kΩ = 1000 Ω |
| megaohm | MΩ | 106 | 1 MΩ = 1,000,000 Ω |
| gigaohm | GΩ | 109 | 1 GΩ = 1,000,000,000 Ω |
| teraohm | TΩ | 1012 | 1 TΩ = 1,000,000,000,000 Ω |
Prefix symbols are case-sensitive. Lowercase m means milli, while uppercase M means mega. One megaohm is one billion times one milliohm. More displayed digits improve presentation of a mathematical result, not the accuracy of the entered component values.
Tolerance, Temperature and Measurement Accuracy
Marked resistor values are nominal. Actual resistance may differ within the component tolerance and may change with temperature, self-heating, age, humidity, mechanical stress and applied voltage. Leads and contacts become important for low resistances, while leakage, surface contamination and meter input resistance matter for high resistances.
Use values measured under the relevant conditions when accuracy matters. Match the number of reported digits to the quality of the input data. Compare a measured network with its expected value using the Percent Error Calculator.
Model Scope and Electrical Safety
The equations apply to ideal linear resistors with the stated connectivity. For a purely resistive sinusoidal AC network, consistent RMS voltage and current values follow the same formulas. Capacitors, inductors, frequency-dependent impedance, phase angle, nonlinear devices and active or negative-resistance elements require another model.
This calculator does not select a resistor rating, wire size, fuse, protection method or safe work procedure. A power result is an ideal dissipation value, not a component recommendation. Confirm voltage, current, pulse energy, duty cycle, derating, temperature rise and applicable standards independently.
Related Calculators
Frequently Asked Questions
What is the formula for resistors in series?
The equivalent resistance is the sum of every resistance: Req = R1 + R2 + ... + Rn. For nonnegative resistors, the total is at least as large as the largest individual resistance.
What is the formula for resistors in parallel?
For positive resistances, 1/Req = 1/R1 + 1/R2 + ... + 1/Rn. Calculate the reciprocal sum, then take its reciprocal to find the equivalent resistance.
How do I calculate two resistors in parallel?
For two positive resistors, use Req = R1R2/(R1 + R2). For example, 100 Ω and 220 Ω in parallel give 68.75 Ω.
Is parallel resistance always smaller than the smallest resistor?
Yes, when two or more branch resistances are positive and finite. Adding a parallel branch increases conductance, so equivalent resistance falls below the smallest individual branch resistance.
What happens if one parallel resistor is zero ohms?
An ideal zero-ohm parallel branch shorts the network, so equivalent resistance is zero. The calculator reports this directly instead of dividing by zero.
Can I mix ohms, kiloohms and megaohms?
Yes. Each resistor has its own unit selector. The calculator converts every value to ohms exactly before applying the selected network formula.
How do I calculate a mixed series-parallel network?
Reduce one clear series or parallel group at a time, replace it with its equivalent resistance, then continue outward. Bridge networks that cannot be reduced this way need a circuit-analysis method such as nodal analysis.
Does resistor order change equivalent resistance?
No, not when the same resistors remain in the same all-series or all-parallel topology. Reordering values does not change a sum or reciprocal sum, but changing connections changes the result.
Does the result include resistor tolerance?
No. The main result uses entered nominal values. Actual resistance may differ because of tolerance, temperature, self-heating, leads, contacts and measurement uncertainty.
Does this calculator work for AC circuits?
It applies to ideal resistors using DC values or consistent RMS values in a purely resistive AC network. Circuits containing capacitance, inductance or phase effects require complex impedance, not resistance alone.
Method and Review Basis
The series, parallel, reduction and distribution relationships were checked against OpenStax series and parallel resistor guidance. Electrical power relationships were checked against OpenStax electric power guidance. The ohm and electrical-unit relationships were checked against NIST electrical-unit guidance and the BIPM SI Brochure. Electrical-work precautions were reviewed against NIOSH electrical safety guidance.
Educational and electrical-safety disclaimer: This tool models ideal passive resistor networks reducible through explicit series and parallel groups. It does not verify a schematic, component rating, installation, protection method, electrical code or safe work condition. Confirm topology, measurements, units, tolerances, thermal limits and standards independently. Only qualified persons should work on or near energized electrical equipment.