Equivalent resistance (Req) is the single resistance value that can replace an entire network of resistors in a circuit without changing the total current drawn from the power supply. When makers and students ask what is R eq, they are looking for the mathematical shortcut that allows you to treat a complex web of series, parallel, or bridge components as one single load. By collapsing a multi-resistor circuit into a single Req value, you can instantly calculate total current, total power dissipation, and overall voltage drop using basic Ohm's Law.
What R Eq Actually Changes in a Real Circuit
Finding the equivalent resistance does not change the physical behavior of the individual components inside the network; a 100Ω resistor still dissipates the same heat whether you calculate it alone or as part of a larger Req. What Req changes is how the power supply perceives the load.
In practical installations, knowing your Req prevents you from undersizing your main fuse or overloading a voltage regulator. If your calculated Req drops too low, your total current spikes, potentially tripping breakers or causing brownouts in sensitive microcontroller circuits.
The Core Formulas: Series, Parallel, and Mixed Networks
To calculate Req, you must break the circuit down into purely series and purely parallel blocks. Here are the foundational formulas you will use on the bench:
Series Networks
When resistors are daisy-chained end-to-end, their resistances simply add up. The current has only one path, so every ohm of resistance stacks directly.
Req = R1 + R2 + R3 + ... + Rn
Parallel Networks
When resistors share the same two nodes, they provide multiple paths for current. The total resistance actually decreases, and the Req will always be lower than the smallest individual resistor in the parallel bank.
1 / Req = (1 / R1) + (1 / R2) + ... + (1 / Rn)
For exactly two resistors in parallel, you can use the faster 'product-over-sum' shortcut:
Req = (R1 × R2) / (R1 + R2)
For a deeper dive into the derivation of these rules, the Electronics Tutorials parallel resistor guide provides excellent schematic breakdowns.
Worked Numeric Example: Calculating a Mixed Resistor Network
Let's move past abstract formulas and calculate Req for a real mixed network. Imagine a 12V DC circuit where Resistor 1 (R1) is in series with a parallel bank made of Resistor 2 (R2) and Resistor 3 (R3).
- R1 = 100Ω
- R2 = 220Ω
- R3 = 330Ω
- Source Voltage (Vs) = 12V DC
Step 1: Collapse the parallel bank.
Using the product-over-sum formula for R2 and R3:
Rparallel = (220 × 330) / (220 + 330)
Rparallel = 72,600 / 550 = 132Ω
Step 2: Add the series resistor.
Now, add R1 to the collapsed parallel value:
Req = 100Ω + 132Ω = 232Ω
Step 3: Calculate total circuit behavior.
With an Req of 232Ω, the total current drawn from the 12V supply is:
Itotal = 12V / 232Ω = 51.72 mA
Where You Meet Equivalent Resistance in Practice
You rarely calculate Req just for academic exercises; it solves specific hardware problems in DIY and professional electronics.
- LED Array Design: When wiring high-power LEDs in series-parallel strings for a 12V automotive system, you must calculate the Req of the entire string to size the main current-limiting resistor and ensure the alternator can handle the total amperage.
- Audio Speaker Matching: Wiring multiple 4Ω or 8Ω speakers in a custom cabinet requires calculating the Req (often called impedance in this context, though technically Z) to ensure you don't drop below the 2Ω minimum threshold of your solid-state amplifier, which would trigger its thermal protection.
- Sensor Voltage Dividers: When using an NTC thermistor in a voltage divider for an ESP32 ADC pin, the parallel Req of the divider network dictates the input impedance seen by the microcontroller's sample-and-hold capacitor. If the Req is too high (typically >10kΩ), the ADC readings will fluctuate wildly due to incomplete capacitor charging.
Common Confusions: R Eq vs. Impedance vs. Internal Resistance
Even experienced hobbyists mix up these three related but distinct concepts. Here is how to keep them straight:
R Eq vs. Impedance (Z)
Equivalent resistance applies strictly to DC circuits or the purely resistive (real) portion of an AC circuit. Impedance (Z) is the AC equivalent that includes reactance—the frequency-dependent resistance introduced by capacitors and inductors. If your network contains a 100µF decoupling capacitor, you cannot calculate a simple DC Req for the whole board; you must calculate impedance at a specific AC frequency.
R Eq vs. Internal Resistance (r)
Req is the equivalent resistance of the external load you built. Internal resistance (lowercase r) is the parasitic resistance inside the power source itself (like the chemical resistance inside a 18650 lithium cell). According to Georgia State University's HyperPhysics, maximum power transfer occurs when your external Req exactly matches the source's internal resistance r. However, in 99% of DIY power supply design, you want your Req to be vastly higher than r to prevent the battery from overheating and sagging in voltage.
Decision Path: Sizing a Single Replacement Resistor
Suppose the mixed resistor network from our numeric example (Req = 232Ω) burned out on a legacy PCB, and you need to replace the entire network with a single physical through-hole resistor. You cannot just buy a '232Ω' resistor, and you must ensure the wattage rating survives the 12V supply. Use this decision tree to pick the exact part.
| Decision Point | Condition / Calculation | Action / Result |
|---|---|---|
| 1. Determine Target Value | Calculated Req is 232Ω. | 232Ω is not a standard E24 value. Nearest are 220Ω and 240Ω. |
| 2. Choose Standard Value | Will a lower resistance (220Ω) harm the circuit? | Lower R increases current. To protect the 12V supply, round UP. Pick 240Ω. |
| 3. Calculate Power Dissipation | P = V² / Req → 144 / 240 | Total power is 0.60 Watts. |
| 4. Apply Derating Rule | Resistors should run at max 50% of rated wattage for longevity. | 0.60W × 2 = 1.2W minimum rating. Pick a 2W package. |
| 5. Select Tolerance & Type | Is this a precision analog sensor circuit? | No, it's a general load. 5% carbon or 1% metal film is fine. Pick 1% Metal Film. |
Frequently Asked Questions
Can R eq be negative?
In passive resistor networks, no. Equivalent resistance is always a positive value. However, in active circuits utilizing op-amps or specialized ICs (like Negative Impedance Converters), you can synthesize a 'negative resistance' that effectively injects current back into the node, canceling out parasitic losses.
Does the physical layout on a breadboard change R eq?
Theoretically, no. Practically, yes. Breadboard contact resistance can add 0.1Ω to 0.5Ω per junction. If you are calculating Req for a high-current, low-voltage circuit (like a 1Ω shunt resistor for current sensing), the breadboard's parasitic resistance will skew your measurements. Solder high-current Req networks directly to perfboard or PCB.
What happens to R eq if one resistor in a parallel bank fails open?
The Req of the parallel bank will instantly increase, because you have lost a current path. In our 220Ω || 330Ω example, if the 220Ω resistor fails open, the bank's resistance jumps from 132Ω to exactly 330Ω. This drops the total circuit current and shifts the voltage drops across the remaining series components.
