# Resistors in Series and Parallel Calculator

> Computes the equivalent resistance of a list of resistors connected all in series (sum) and all in parallel (reciprocal of the sum of reciprocals), plus the parallel conductance.

- Calculator id: `resistors-series-parallel` · Category: Physics (`physics`) · Tool name: `calculate_resistors_series_parallel`
- Canonical page: https://tttkmbb.com/physics/resistors-series-parallel · This document: https://tttkmbb.com/physics/resistors-series-parallel.md · JSON definition: https://tttkmbb.com/physics/resistors-series-parallel.json

## Purpose

Computes the equivalent resistance of a list of resistors connected all in series (sum) and all in parallel (reciprocal of the sum of reciprocals), plus the parallel conductance.

**Use when:** You have two or more resistor values and need the total resistance if they are wired in series or in parallel, or want to check a parallel combination against a target value.

**Do not use when:** The network mixes series and parallel branches (reduce it group by group with repeated calls), the components are capacitors or inductors, or you need voltage, current or power for one resistor (use ohms-law).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `resistances_ohm` | number_list | Ω | required | Resistor values in ohms, comma-separated (1 kΩ = 1000). All values must be positive. |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `resistor_count` | integer |  | How many values were combined. |
| `series_total_ohm` | number | Ω | R1 + R2 + … (always larger than the largest resistor). |
| `parallel_total_ohm` | number | Ω | 1 / (1/R1 + 1/R2 + …) (always smaller than the smallest resistor). |
| `parallel_conductance_s` | number | S | Sum of conductances 1/R1 + 1/R2 + … in siemens. |

## Formula

`series_total_ohm = R1 + R2 + …; 1 / parallel_total_ohm = 1/R1 + 1/R2 + …`

## Data Sources

- HyperPhysics – Resistance and resistor combinations — http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html (reference, retrieved 2026-09-23)
- Wikipedia – Series and parallel circuits — https://en.wikipedia.org/wiki/Series_and_parallel_circuits (reference, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/resistors-series-parallel?resistances_ohm=…`
- `POST https://tttkmbb.com/api/v1/calculate/resistors-series-parallel` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/resistors-series-parallel · OpenAPI operationId `calculate_resistors_series_parallel` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "resistors-series-parallel", "inputs": {…}}`

## Example

- 100, 220 and 470 Ω: inputs `{"resistances_ohm":[100,220,470]}` → `{"resistor_count":3,"series_total_ohm":790,"parallel_total_ohm":59.9768,"parallel_conductance_s":0.016673}`
- Two 10 Ω resistors: inputs `{"resistances_ohm":[10,10]}` → `{"resistor_count":2,"series_total_ohm":20,"parallel_total_ohm":5}`

```
GET https://tttkmbb.com/api/v1/calculate/resistors-series-parallel?resistances_ohm=100%2C220%2C470
```

## Limitations

The network mixes series and parallel branches (reduce it group by group with repeated calls), the components are capacitors or inductors, or you need voltage, current or power for one resistor (use ohms-law). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**How do I handle a mixed circuit?**

Reduce it step by step: combine each purely parallel or purely series group into one equivalent resistor, then combine the results, calling this calculator for each group.

**Why is the parallel total less than the smallest resistor?**

Each added parallel path lets more current flow at the same voltage, increasing total conductance; n equal resistors R in parallel give R / n.

## Related

- [Ohm's Law Calculator](https://tttkmbb.com/physics/ohms-law.md) — Voltage, current and power for the equivalent resistance.
- [Electric Power Calculator](https://tttkmbb.com/physics/electric-power.md) — Power and energy drawn by the circuit.