When you are analyzing a circuit on the bench or designing a voltage divider, you rarely deal with just one resistor. You deal with networks. The Req formula (Equivalent Resistance formula) is the mathematical tool that collapses a complex web of series and parallel resistors into a single, equivalent component value. This single value tells you exactly how much current the entire network will draw from your power supply.
Below is the definitive guide to the Req formula, including exact symbol definitions, rearranged algebraic forms for finding unknown components, and fully worked examples with strict unit tracking.
The Core Req Formula and Symbol Definitions
There is no single "master" Req formula; rather, the formula changes based on the topology of the circuit. Here are the three foundational equations you will use 99% of the time.
1. Series Network
Req = R1 + R2 + ... + Rn
2. Parallel Network (General)
1 / Req = 1 / R1 + 1 / R2 + ... + 1 / Rn
3. Parallel Network (Two Resistors Only / Product-over-Sum)
Req = (R1 × R2) / (R1 + R2)
Symbol Definition Table
| Symbol | Definition | Standard Unit |
|---|---|---|
| Req | Equivalent Resistance of the entire network | Ohms (Ω) |
| R1, R2 | Resistance of individual components in the network | Ohms (Ω) |
| Rn | Resistance of the n-th component in the network | Ohms (Ω) |
| n | Total number of resistors in the specific series/parallel branch | Unitless (integer) |
When the Req Formula Applies (and When It Breaks)
The Req formula is derived directly from Ohm's Law and Kirchhoff's Circuit Laws. However, it relies on specific physical assumptions. If you violate these assumptions, your calculated Req will not match what your multimeter reads on the bench.
Core Assumptions
- Linearity: The resistors must be Ohmic. Their resistance must remain constant regardless of the voltage applied across them or the current flowing through them. (This excludes varistors and thermistors).
- Steady-State DC or Low-Frequency AC: At high frequencies (typically above 100 kHz), the parasitic inductance of the component leads and the parasitic capacitance between the resistive film tracks begin to dominate. A 10 kΩ carbon film resistor might exhibit an impedance of 8 kΩ at 10 MHz. The basic Req formula ignores these reactive parasitics.
- Constant Temperature: Resistors have a Temperature Coefficient of Resistance (TCR). If a resistor is dissipating heavy power (e.g., a wirewound power resistor running at 5W), it will self-heat, shifting its actual resistance away from the nominal calculated Req.
The Unit Mistake That Breaks Calculations
The most common mistake hobbyists make is mixing base units and prefixes without converting. If you have a 4.7 kΩ resistor in series with a 220 Ω resistor, you cannot calculate Req as 4.7 + 220 = 224.7. You must convert all values to base Ohms first: 4700 Ω + 220 Ω = 4920 Ω (or 4.92 kΩ). Always track your units through the algebra.
Realistic Answer Magnitudes
When you calculate Req, does the answer make physical sense? Here is what realistic magnitudes look like in practical electronics:
- 10 mΩ to 100 mΩ: Current sensing shunt networks (e.g., parallel bank of milliohm surface-mount resistors).
- 100 Ω to 10 kΩ: Standard signal paths, LED current limiters, and voltage dividers for microcontroller ADCs.
- 100 kΩ to 1 MΩ: I2C pull-up resistor networks, high-impedance bias networks, and oscilloscope probe compensation circuits.
Rearranged Forms: Solving for Unknowns
On the bench, you rarely just calculate Req from known parts. Usually, you have a target Req and need to figure out what missing resistor to add. Here are the algebraically rearranged forms of the Req formula.
- Find an unknown series resistor (Rx):
Rx = Req - Rknown_total - Find an unknown parallel resistor (Rx) to pair with a known resistor (R1) to achieve a target Req:
Rx = (Req × R1) / (R1 - Req) - Find the required number of identical resistors (n) in parallel to achieve a target Req:
n = Rsingle / Req
Worked Examples with Unit Tracking
Let's apply these formulas to real-world scenarios, paying strict attention to intermediate steps and unit cancellation.
Problem 1: Mixed Series-Parallel Network
Scenario: You have a circuit where R1 (100 Ω) is in series with a parallel bank consisting of R2 (300 Ω) and R3 (600 Ω). What is the total Req?
Step 1: Collapse the parallel bank.
Use the product-over-sum formula for R2 and R3.
Rparallel = (R2 × R3) / (R2 + R3)
Rparallel = (300 Ω × 600 Ω) / (300 Ω + 600 Ω)
Rparallel = 180,000 Ω² / 900 Ω
Rparallel = 200 Ω
Step 2: Add the series component.
Now treat the 200 Ω parallel bank as a single resistor in series with R1.
Req = R1 + Rparallel
Req = 100 Ω + 200 Ω
Req = 300 Ω
Problem 2: Reverse-Engineering a Parallel Resistor
Scenario: You are building a custom sensor bridge and need an exact equivalent resistance of 150 Ω. You only have a 220 Ω metal film resistor in your bin. What value resistor (Rx) must you place in parallel with the 220 Ω resistor to hit your 150 Ω target?
Step 1: Select the rearranged parallel formula.
Rx = (Req × R1) / (R1 - Req)
Step 2: Substitute values and track units.
Rx = (150 Ω × 220 Ω) / (220 Ω - 150 Ω)
Rx = 33,000 Ω² / 70 Ω
Rx = 471.42 Ω
Bench Application: 471.42 Ω is not a standard E24 resistor value. You would select a standard 470 Ω 1% tolerance resistor. Let's verify the actual Req with the 470 Ω part:
Req_actual = (220 × 470) / (220 + 470) = 103,400 / 690 = 149.85 Ω.
This is well within the 1% tolerance margin for precision analog circuits.
Frequently Asked Questions
How do I calculate the req formula for 3 or more resistors in parallel?
You cannot use the simple product-over-sum formula for three resistors. You must use the reciprocal sum formula: 1/Req = 1/R1 + 1/R2 + 1/R3. Calculate the right side of the equation as a decimal or common fraction, and then take the reciprocal (1 divided by your answer) to find Req. Alternatively, you can collapse them two at a time: find the equivalent of R1 and R2 using the product-over-sum method, and then take that result and use the product-over-sum method again with R3.
What is the shortcut req formula for identical resistors in parallel?
If you have n identical resistors (all the exact same ohm value, R) wired in parallel, the formula simplifies dramatically to: Req = R / n. For example, if you put four 100 Ω resistors in parallel, the Req is simply 100 / 4 = 25 Ω. This is a common trick used by engineers to increase the total power dissipation rating of a resistor network while achieving a lower target resistance.
Why is the parallel req always smaller than the smallest resistor in the group?
Think of resistance as the restriction of flow, and its inverse, conductance (G = 1/R), as the ease of flow. When you add resistors in parallel, you are adding new, separate pathways for current to flow. Even if you add a massive 1 MΩ resistor in parallel with a tiny 10 Ω resistor, that 1 MΩ path still allows a tiny bit of extra current to bypass the 10 Ω path. Because total current flow increases for the same applied voltage, the overall equivalent resistance must decrease. Therefore, Req will always be strictly less than the smallest individual resistor in the parallel bank.
Does the req formula work for AC impedance?
Yes, but the math shifts from scalar algebra to complex vector math. In AC circuits containing capacitors and inductors, we use the Zeq formula (Equivalent Impedance). The topology rules remain exactly the same (series impedances add directly, parallel impedances add reciprocally), but you must track the real (resistive) and imaginary (reactive) components using complex numbers (e.g., Z = R + jX). For further reading on foundational DC network theorems, refer to the HyperPhysics series and parallel circuit documentation or the Electronics Tutorials parallel resistor guide.
