Find equivalent resistance for resistors in parallel or series — with step-by-step breakdown, conductance, current, and power.
Resistors connected in parallel share the same two nodes, meaning the same voltage appears across every branch while the current divides among the branches. The single most important property of a parallel combination is that the equivalent resistance is always less than the smallest individual resistor — adding any parallel branch always provides an additional current path, reducing the total opposition to current flow. Engineers use parallel resistors constantly: to create non-standard resistance values from E12-series components, to increase the power-handling capacity of a network (multiple 1 W resistors sharing load instead of one larger one), to set bias currents in transistor amplifiers, and to build precision voltage dividers and filters. Hobbyists encounter parallel resistors whenever two components share a power rail or whenever they need a value not found in a standard parts kit. This calculator handles the core formula — 1/R_eq = 1/R₁ + 1/R₂ + … + 1/Rₙ — for up to 10 resistors simultaneously. Every resistor row has its own unit selector (µΩ, mΩ, Ω, kΩ, MΩ, GΩ), so you can freely mix milliohm shunts with megaohm bias resistors in a single calculation. Results are auto-scaled to the most readable unit: a 0.00047 Ω answer displays as 0.47 mΩ, while 1,500,000 Ω displays as 1.5 MΩ. Beyond the basic equivalent-resistance output, the tool offers a Missing Resistor Solver: enter the target R_eq and all-but-one known resistors, and the calculator algebraically solves 1/Rₙ = 1/R_eq − 1/R₁ − … − 1/Rₙ₋₁ to find the unknown value. This is handy when you need a specific resistance and want to know which second resistor to add in parallel to hit the target. For power-electronics work, the optional Supply Voltage field unlocks current distribution and power dissipation per branch. Because voltage is identical across all parallel branches, the current through each branch is simply I = V/R, and the power is P = V²/R. The calculator sums branch currents to give total current and total power, letting you verify that no single component exceeds its rated wattage. A companion Series Mode tab lets you calculate series resistance (R_total = R₁ + R₂ + … + Rₙ) using the same dynamic row interface, making it easy to switch between configurations without re-entering values. The step-by-step calculation panel walks through the reciprocal method explicitly: it lists 1/R for each branch, sums them into G_total (total conductance in Siemens), then takes the final reciprocal to give R_eq. Conductance is displayed alongside resistance because in parallel circuits the conductances simply add — a fact that makes the math intuitive and connects to broader circuit-analysis techniques like nodal analysis. Visual aids include a pure-SVG schematic of the parallel circuit (resistor symbols on horizontal branches between vertical bus bars), a horizontal bar chart comparing each Rₙ against R_eq, and — when a supply voltage is entered — a current-distribution bar chart highlighting which branches carry the most current. E12-series preset dropdowns on every row let you snap to standard values instantly, and six quick-fill buttons populate R₁ with common values (10 Ω, 100 Ω, 1 kΩ, 4.7 kΩ, 10 kΩ, 47 kΩ) for rapid experimentation. Results can be copied to the clipboard with one click or exported as a CSV file containing all branch values, conductances, currents, and power readings — useful for documentation in design reports or spreadsheets.
Understanding Parallel Resistors
What Are Parallel Resistors?
Two or more resistors are in parallel when both terminals of each resistor are connected to the same two nodes of a circuit. This arrangement means every resistor experiences exactly the same voltage across its terminals. Unlike a series connection — where resistors share the same current — a parallel connection shares the same voltage and splits the current. The total current flowing into the parallel combination equals the sum of the individual branch currents: I_total = I₁ + I₂ + … + Iₙ. Because each additional parallel branch gives current another route through the circuit, the equivalent (combined) resistance decreases every time you add a parallel resistor. Even adding a very large resistor in parallel will slightly lower the total resistance, though the effect diminishes as the added value grows.
How Is Equivalent Resistance Calculated?
The standard formula uses the reciprocal (inverse) method: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ, therefore R_eq = 1 ÷ (1/R₁ + 1/R₂ + … + 1/Rₙ). For just two resistors there is a convenient shortcut — the product-over-sum formula: R_eq = (R₁ × R₂) / (R₁ + R₂). For N identical resistors each of value R, the result simplifies to R/N. The conductance view offers an alternative perspective: conductance G = 1/R (measured in Siemens), so parallel conductances simply add — G_total = G₁ + G₂ + … + Gₙ — and R_eq = 1/G_total. All units must be converted to Ohms before applying these formulas; the calculator handles µΩ, mΩ, kΩ, MΩ, and GΩ conversions automatically.
Why Does Parallel Resistance Matter?
Understanding parallel resistance is fundamental to circuit design. It explains why adding more load devices to a power supply decreases the total load resistance and draws more current. It is the basis for calculating equivalent impedances in AC filter networks, understanding how LED arrays distribute current, and designing precision resistor networks that hit non-standard values. In safety engineering, parallel paths through the human body lower tissue resistance and determine shock severity. For audio engineers, speaker impedance in a PA system changes dramatically when cabinets are wired in parallel. Power electronics designers use parallel resistors to spread heat across multiple components, reducing thermal stress. Every electrical and electronics engineer needs this calculation in their toolbox.
Limitations and Practical Considerations
This calculator assumes ideal, purely resistive components. Real-world resistors have tolerance (typically ±1% to ±5%), which means the actual equivalent resistance will differ slightly from the computed value. At high frequencies, parasitic inductance and capacitance of physical resistors become significant and the simple DC formula no longer applies. If one resistor is 0 Ω (a short circuit), the equivalent resistance is 0 Ω regardless of other resistors — the tool reports this correctly. For non-linear devices (diodes, transistors, thermistors) Ohm's law and simple resistance formulas do not apply. Power ratings are critical in practice: each branch carries P = V²/R watts; a resistor must not be operated above its rated wattage. Always verify power dissipation using the Supply Voltage field before committing to a design.
Formulas Used
Parallel Resistance (General)
R_eq = 1 / (1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ)
The reciprocal of the equivalent resistance equals the sum of the reciprocals of all individual resistances.
Two-Resistor Shortcut (Product over Sum)
R_eq = (R₁ × R₂) / (R₁ + R₂)
Applies only to exactly two resistors in parallel. Faster than the general reciprocal method for the common two-resistor case.
Missing Resistor
Rₙ = 1 / (1/R_eq − 1/R₁ − 1/R₂ − … − 1/Rₙ₋₁)
Algebraic rearrangement to solve for one unknown resistor given the desired equivalent resistance and all other resistors.
N Identical Resistors
R_eq = R / N
When all N parallel resistors have the same value R, the equivalent resistance is simply R divided by N.
Conductance Method
G_total = G₁ + G₂ + … + Gₙ, where G = 1/R (Siemens)
Conductances add directly in parallel. R_eq = 1 / G_total.
Branch Current (given supply voltage V)
I_n = V / Rₙ
Voltage is equal across all parallel branches. Each branch current follows Ohm's Law independently.
Power Dissipation Per Branch
P_n = V² / Rₙ
Power dissipated in each branch resistor given supply voltage V. Total power P_total = V² / R_eq.
Series Resistance
R_total = R₁ + R₂ + R₃ + … + Rₙ
For the companion series mode: resistances add directly when connected end-to-end.
Reference: E12 Standard Resistor Series
E12 Series Values (Base Decade)
The E12 series contains 12 values per decade spaced roughly 20% apart. Multiply by powers of 10 (10×, 100×, 1 k×, 10 k×, etc.) to get the full range of standard resistor values available from most component suppliers.
| E12 Value (Ω) | Approx. Spacing |
|---|---|
| 10 | — |
| 12 | +20% |
| 15 | +25% |
| 18 | +20% |
| 22 | +22% |
| 27 | +23% |
| 33 | +22% |
| 39 | +18% |
| 47 | +21% |
| 56 | +19% |
| 68 | +21% |
| 82 | +21% |
Worked Examples
Two Identical 10 Ω Resistors in Parallel
R₁ = 10 Ω, R₂ = 10 Ω
1/R_eq = 1/10 + 1/10 = 0.1 + 0.1 = 0.2 S
R_eq = 1 / 0.2 = 5 Ω
R_eq = 5 Ω — exactly half the individual value, as expected for two identical parallel resistors.
Three Different Resistors: 10 Ω, 5 Ω, 2 Ω
R₁ = 10 Ω, R₂ = 5 Ω, R₃ = 2 Ω
1/R_eq = 1/10 + 1/5 + 1/2 = 0.1 + 0.2 + 0.5 = 0.8 S
R_eq = 1 / 0.8 = 1.25 Ω
R_eq = 1.25 Ω — less than the smallest resistor (2 Ω), confirming the key parallel rule.
Find Missing Resistor for 75 Ω Target
Desired R_eq = 75 Ω, R₁ = 100 Ω (known)
1/R₂ = 1/75 − 1/100
1/R₂ = 0.01333 − 0.01000 = 0.00333 S
R₂ = 1 / 0.00333 = 300 Ω
R₂ = 300 Ω — add a 300 Ω resistor in parallel with the 100 Ω to achieve exactly 75 Ω equivalent.
How to Use the Parallel Resistor Calculator
Enter Your Resistor Values
Type the value of each resistor into the numbered input fields (R1, R2, …). Use the unit dropdown beside each field to choose the correct unit (Ω, kΩ, MΩ, etc.). You can also click a quick-fill button to pre-populate R1 with a common E12 value, or use the E12 dropdown on any row to snap that row to a standard resistor value.
Add or Remove Resistors
Click 'Add Resistor' to add up to 10 resistors. Use the trash icon on any row to remove it. The calculation updates automatically as you type — there is no separate Calculate button needed. Switch between the Parallel and Series tabs at the top to change the connection mode.
Optional: Voltage, Mode, and Precision
Enter a supply voltage to reveal current and power dissipation per branch — useful for verifying component power ratings. Switch Calculation Mode to 'Find Missing Resistor' if you know the desired R_eq and need to find the value of one unknown resistor. Adjust Significant Figures to control the display precision.
Review Results and Export
The right panel shows the equivalent resistance, conductance, a visual circuit schematic, a resistance comparison bar chart, step-by-step reciprocal math, and (if a voltage is entered) current and power per branch. Copy the result to clipboard, export all data as a CSV file, or print the results page for documentation.
Frequently Asked Questions
Why is the parallel resistance always less than the smallest resistor?
Adding a resistor in parallel always provides an additional path for current to flow. Even if the new resistor is very large (high resistance), it still carries some extra current, which means the total resistance the source sees must be lower. Mathematically, the reciprocal sum 1/R_eq = 1/R₁ + 1/R₂ + … grows with every term added, so R_eq = 1/(sum) shrinks. The limiting case is two identical resistors of R: their equivalent is R/2 — exactly half. For N identical resistors, the equivalent is R/N. No parallel combination can ever equal or exceed the smallest individual resistor.
What is the product-over-sum formula and when should I use it?
The product-over-sum formula R_eq = (R₁ × R₂) / (R₁ + R₂) is a convenient shortcut for exactly two resistors in parallel. It avoids the two-step reciprocal calculation: you multiply the two values, divide by their sum, and get the result directly. It is only valid for exactly two resistors — for three or more you must use the full reciprocal method (1/R_eq = 1/R₁ + 1/R₂ + 1/R₃). A common mistake is applying the product-over-sum to three resistors by chaining it: first combine R₁ and R₂ into R₁₂, then combine R₁₂ with R₃ — this is mathematically correct since the operation is associative.
How do I find a missing resistor for a desired equivalent resistance?
Switch the Calculation Mode to 'Find Missing Resistor', enter the desired R_eq in the Desired Total Resistance field, and enter all known resistors in the row inputs. The calculator solves 1/Rₙ = 1/R_eq − 1/R₁ − 1/R₂ − … algebraically. One important constraint: for a solution to exist, R_eq must be strictly less than every individual known resistor — if it is not, the calculator will show an 'impossible' error. In practice this mode is useful when you have a target impedance (such as 75 Ω for a video termination) and need to know which resistor to add in parallel with an existing one.
What are conductance and Siemens?
Conductance (G) is the reciprocal of resistance: G = 1/R. Its SI unit is the Siemens (S), also written as mho (℧). Conductance measures how easily current flows through a component — a high conductance means low resistance and vice versa. In a parallel circuit, conductances add directly: G_total = G₁ + G₂ + … + Gₙ, which is why the reciprocal method works. Thinking in conductance can be more intuitive for parallel circuits: adding a 1 S conductor in parallel with a 2 S conductor gives 3 S total, which is R_eq = 1/3 Ω ≈ 0.333 Ω. The conductance view is shown below the main result alongside each branch value.
Why does voltage matter for parallel resistors?
In a parallel circuit, voltage is identical across every branch — it is a defining property of parallel connections (whereas series circuits share the same current). Knowing the supply voltage unlocks the current and power calculations: I_branch = V / R_branch (Ohm's Law), and P_branch = V² / R_branch. These values matter practically: you need to verify that each resistor's wattage rating is not exceeded. For example, a 10 kΩ resistor at 12 V dissipates P = 144/10,000 = 14.4 mW, well within a standard ¼ W rating. But a 100 Ω resistor at 12 V dissipates 1.44 W — exceeding a ¼ W resistor and requiring a 2 W component.
Does the same formula apply to inductors and capacitors?
Yes — with an important twist. Inductors in parallel follow the same reciprocal formula as resistors in parallel: 1/L_eq = 1/L₁ + 1/L₂ + … (assuming no mutual coupling between inductors). Capacitors, however, are the mathematical dual: capacitors in series follow the reciprocal formula (1/C_eq = 1/C₁ + 1/C₂ + …), and capacitors in parallel simply add (C_eq = C₁ + C₂ + …). This cross-domain analogy is highlighted in the calculator results panel. It arises from the duality of voltage and current in circuit theory and is a useful mnemonic: parallel resistors/inductors use reciprocals; parallel capacitors add directly.