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Parallel Resistance Calculator

Calculate the total equivalent resistance of two to four resistors in parallel - always less than the smallest individual resistor in the network.

Frequently Asked Questions

Is 1/R<sub>parallel</sub> always less than 1/R<sub>smallest</sub>?

Yes, without exception. The total conductance of a parallel combination (1/Rparallel) equals the sum of the individual conductances (1/R1 + 1/R2 + ...). Since every added term is positive, the total conductance is strictly greater than any single term alone. This means the total parallel resistance is strictly less than any individual resistance in the combination. Even adding a 1 MΩ resistor in parallel with a 100 Ω resistor reduces the total resistance slightly (from 100 Ω to 99.99 Ω). The reduction is negligible when one resistor is much larger than the others, but it is always real and always in the direction of lower total resistance.

How do I simplify a resistor ladder network?

A resistor ladder (common in R-2R DAC networks and transmission line models) is simplified by starting at the far end and working back toward the source. At the last rung, the two resistors form a simple series or parallel combination. Replace them with their equivalent, then treat the previous rung with its series or parallel relationship to the new equivalent. Continue until only the source-side input remains. For an R-2R ladder specifically, each step looking from the right produces the same equivalent resistance R, which is why R-2R ladders are used in DAC circuits: the impedance is uniform regardless of the number of rungs, making the network easy to analyze and scale.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

Resistors in parallel share the same voltage but split the current among themselves. The total resistance is found by adding the reciprocals of each resistance and then taking the reciprocal of that sum. The combined resistance is always smaller than the smallest individual resistor, because adding more paths gives current more ways to flow.

How To Use It

Enter at least two resistor values in ohms. The third and fourth fields are optional and are included only when you fill them in with a positive value. The calculator returns the total equivalent resistance of the parallel combination, which you can then treat as a single resistor in the rest of your circuit.

Why Parallel Lowers Resistance

Each added branch opens another channel for current at the same voltage, so the network conducts more easily. Two equal resistors in parallel give exactly half their value. This is the opposite of series resistors, which simply add up and always increase the total.

Tips

The result is always below the smallest resistor.
Leave the optional fields blank for a two-resistor network.
Equal parallel resistors halve, third, or quarter the value.

Why Parallel Resistance Matters in Circuit Design

When two or more resistors share the same two nodes in a circuit, they are in parallel. By Kirchhoff's voltage law, the voltage across each resistor must be identical (both ends are connected to the same nodes). But by Kirchhoff's current law, the total current entering the node equals the sum of the individual branch currents. More parallel branches mean more total current flows for the same voltage, which means the network behaves like a lower resistance. This is why parallel resistance always produces a result smaller than the smallest individual value.

The output impedance of a parallel combination determines how stiff the node is: a lower Thevenin resistance means the node voltage changes less when current is drawn. This principle governs how current sources, amplifier outputs, and power rails behave under varying loads.

Current Divider Rule

When a known total current I_total enters a parallel resistor network, the current through any individual branch n follows the current divider rule: I_n = I_total x (R_parallel / R_n), where R_parallel is the total parallel resistance of all branches. A branch with a smaller resistance carries a proportionally larger share of the current. For two equal resistors, each carries exactly half the total current. The current divider rule is the dual of the voltage divider rule for series resistors and is equally useful in circuit analysis.

Norton Equivalent

Just as the Thevenin theorem simplifies any resistive network into a series voltage source and resistance, the Norton theorem simplifies it into a parallel current source and resistance. The Norton current I_N is the short-circuit current at the output terminals, and the Norton resistance R_N is the same as the Thevenin resistance (R_N = V_th / I_N). The Norton equivalent is especially convenient when analyzing parallel circuits and current-driven nodes, such as transistor output stages and current mirror loads in analog circuits.

Practical Parallel Resistance Applications

Parallel resistance combinations appear throughout practical circuit design, both for achieving specific values and for managing power dissipation.

Precision Resistance Values

Standard resistor series (E12, E24, E96) provide a defined set of values, and not every desired resistance is available. By combining two standard-value resistors in parallel, designers can achieve values that fall between standard steps. For example, 10 kΩ in parallel with 10 kΩ gives exactly 5 kΩ, a value not in the E12 series. More complex combinations allow targeting almost any resistance within the tolerance of the individual components. This technique is used in precision analog circuits, filter designs, and reference voltage generation where exact resistor values matter but special-order parts are impractical.

Power Distribution

When a circuit requires more power dissipation than a single resistor of a given value can handle, paralleling two or more resistors of twice the required resistance splits the power equally between them. For example, if 2 W must be dissipated in a 100 Ω circuit element, using two 200 Ω 1 W resistors in parallel achieves 100 Ω equivalent resistance with each resistor dissipating exactly 1 W. Motor braking resistor banks in industrial variable frequency drives, load resistors in high-power RF amplifiers, and dummy loads for power supply testing all use parallel power resistors to manage heat dissipation while maintaining the required resistance value.

Transmission Line Termination

Signal transmission lines (coaxial cable, PCB microstrip) must be terminated in their characteristic impedance to prevent reflections. When standard resistor values do not match the required impedance, parallel combinations fill the gap. A 75 Ω video termination can be built from two 150 Ω resistors in parallel when only 150 Ω resistors are available. A 50 Ω RF load is easily constructed from two 100 Ω resistors, both of which are widely available in surface-mount form. For precision applications, the resistors should be chosen for low inductance (chip resistors rather than wirewound) and adequate power rating for the signal levels involved.

Resistor Network Simplification

Complex resistor networks that cannot be reduced by simple series and parallel steps require more advanced analysis techniques. Recognizing when a network is reducible and when it requires special methods saves significant analysis time.

The basic strategy for series-parallel simplification is to work from the inside out: find innermost groups of resistors that are clearly in series or clearly in parallel, combine them into a single equivalent resistance, and repeat until a single equivalent resistance remains. This works for most ladder networks and for many real-world circuits.

Bridge Network (Wheatstone)

A Wheatstone bridge network cannot be reduced by simple series or parallel combinations because no two resistors share exactly two nodes without additional connections. To find the equivalent resistance of a bridge, either apply the star-delta (Y-Delta or Y-Δ) transformation to convert the bridge into a reducible topology, or use node voltage analysis (Kirchhoff's current law at each node) to solve directly. SPICE circuit simulators handle arbitrary networks instantly by matrix methods, making them the practical tool for complex networks in real designs.

Superposition

For networks with multiple voltage or current sources, the superposition theorem simplifies analysis. Each independent source is treated one at a time: all other voltage sources are replaced with short circuits and all other current sources are replaced with open circuits. The response due to each source alone is calculated, then all the individual responses are summed to find the total. Superposition is valid for linear networks only and does not work for nonlinear elements like diodes or transistors operating in their nonlinear regions.

Real Resistor Tolerances

Two nominal 10 kΩ resistors in parallel should give exactly 5 kΩ, but real resistors have manufacturing tolerances. With ±5% tolerance parts, each resistor could be anywhere from 9,500 Ω to 10,500 Ω. The worst-case parallel combination ranges from (9,500 || 9,500) = 4,750 Ω to (10,500 || 10,500) = 5,250 Ω, a total tolerance of ±5% on the result. This is not surprising: the parallel combination inherits the tolerance of its constituent resistors. For precision applications, use 1% or 0.1% tolerance resistors and verify that temperature coefficients are matched to avoid ratio drift over temperature.