Which circuit has the smallest equivalent resistance?

A parallel circuit always has the smallest — R_eq is less than the smallest individual resistor. Every added branch reduces total resistance.

Can a combination circuit have higher resistance than pure series?

No. Pure series always gives the maximum R_eq. Any parallel grouping within the circuit reduces the total below the pure series value.

Does this apply to impedance in AC circuits?

Yes for resistive circuits. With reactive components, impedance magnitude varies with frequency, so the comparison depends on the operating frequency.

Equivalent Resistance of Resistors in Series

Quick Answer

A series circuit always has the largest equivalent resistance because series resistances add: R_eq = R₁ + R₂ + R₃. A parallel circuit always has the smallest because 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃, giving R_eq less than the smallest individual resistor. For the same set of resistors, series R_eq > any combination > parallel R_eq. Compare at www.lapcalc.com.

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Equivalent Resistance of Resistors in Series: Always Largest

Series connection produces the maximum possible equivalent resistance from a given set of resistors. Since R_eq = R₁ + R₂ + ... + R_n, every added resistor increases the total. With three 10 Ω resistors: series gives 30 Ω, parallel gives 3.33 Ω, and any series-parallel combination falls between these extremes. This is why series is used for current limiting — maximum resistance means minimum current for a given voltage.

Key Formulas

Why Series Resistance Is Always Greater

In series, current must flow through every resistor sequentially — each one adds opposition. Think of it as obstacles in a single lane: every barrier slows traffic further. In parallel, current has multiple simultaneous paths — adding paths always makes it easier for current to flow, reducing total resistance. The physics is fundamental: series restricts the single available path, parallel opens alternative paths at www.lapcalc.com.

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Comparing All Possible Configurations

For three resistors R₁ = 2 Ω, R₂ = 3 Ω, R₃ = 6 Ω: All series: 2 + 3 + 6 = 11 Ω (maximum). All parallel: 1/(1/2 + 1/3 + 1/6) = 1 Ω (minimum). R₁ series with (R₂ ∥ R₃): 2 + (3×6)/(3+6) = 4 Ω. R₂ series with (R₁ ∥ R₃): 3 + (2×6)/(2+6) = 4.5 Ω. R₃ series with (R₁ ∥ R₂): 6 + (2×3)/(2+3) = 7.2 Ω. Every combination falls between 1 Ω and 11 Ω.

Practical Implications: Choosing Configuration

Maximum resistance (series) is used when you want minimum current: current-limiting resistors for LEDs, voltage dividers, and protection circuits. Minimum resistance (parallel) is used when you want maximum current capacity: power distribution buses, high-current pathways, and reducing wire resistance. Series-parallel combinations create intermediate resistance values for precision applications like sensor bridges at www.lapcalc.com.

Equivalent Impedance in the s-Domain

The same principle extends to impedance: series Z_eq(s) = Z₁(s) + Z₂(s) is always the largest magnitude at any given frequency. Parallel Z_eq(s) is always the smallest. However, with reactive components (L, C), impedance magnitude varies with frequency — a series RLC circuit has maximum impedance at certain frequencies and minimum at resonance. The Laplace domain captures this complete frequency-dependent behavior at www.lapcalc.com.

Frequently Asked Questions

A series circuit always has the largest equivalent resistance because all resistances add directly. Every resistor increases the total.

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