To convert a specific circuit resistance of 10 Ω to conductance, the direct answer is 0.1 Siemens (S). The formula used is G = 1 / R, substituting the values as G = 1 / 10 Ω = 0.1 S. If your goal is to convert a material's bulk resistivity to conductivity—such as converting annealed copper's resistivity of 1.68 × 10⁻⁸ Ω·m to conductivity—the answer is 5.96 × 10⁷ S/m using the formula σ = 1 / ρ. Whether you are sizing a shunt resistor or calculating trace widths on a PCB, knowing how to flip between opposition to current and facilitation of current is a fundamental bench skill.
The Core Formulas: Circuit Conductance vs Material Conductivity
Before pulling out a calculator, you must define whether you are working at the circuit level or the material level. The terms are often conflated in casual shop talk, but they represent entirely different physical properties.
1. Circuit Level (Resistance to Conductance)
Conductance (G) measures how easily a specific component or circuit allows current to flow. It is the exact mathematical reciprocal of resistance (R). The standard unit is the Siemens (S), though older schematics and vintage texts from the mid-20th century often use the 'mho' (℧)—which is simply 'ohm' spelled backward with an inverted Omega symbol.
- Formula: G = 1 / R
- Units: Siemens (S) = 1 / Ohms (Ω)
- Example: A 50 Ω braking resistor has a conductance of 1 / 50 = 0.02 S (or 20 mS).
2. Material Level (Resistivity to Conductivity)
Conductivity (σ) is an intrinsic property of a material, independent of its shape or size. It is the reciprocal of resistivity (ρ). This is the value you look up when comparing 6061 aluminum busbars against C11000 copper busbars.
- Formula: σ = 1 / ρ
- Units: Siemens per meter (S/m) = 1 / (Ohm-meters (Ω·m))
- Example: Pure aluminum has a resistivity of roughly 2.65 × 10⁻⁸ Ω·m. Its conductivity is 1 / (2.65 × 10⁻⁸) ≈ 3.77 × 10⁷ S/m.
For a deep dive into the historical shift from mhos to Siemens, the All About Circuits textbook chapter on conductance provides excellent context on why the SI system standardized the Siemens in 1971.
Resistance to Conductance Reference Table (±20% Range)
When tuning a current-sense circuit or selecting parallel shunt resistors, it is helpful to see how small shifts in resistance alter the total conductance. The table below uses a baseline of 10 Ω and maps a ±20% tolerance range. Notice that the relationship is non-linear: a 20% drop in resistance yields a 25% increase in conductance.
| Resistance (Ω) | Conductance (S) | Conductance (mS) | Delta from Baseline |
|---|---|---|---|
| 8.0 Ω (-20%) | 0.1250 S | 125.0 mS | +25.0% |
| 9.0 Ω (-10%) | 0.1111 S | 111.1 mS | +11.1% |
| 10.0 Ω (Baseline) | 0.1000 S | 100.0 mS | 0.0% |
| 11.0 Ω (+10%) | 0.0909 S | 90.9 mS | -9.1% |
| 12.0 Ω (+20%) | 0.0833 S | 83.3 mS | -16.7% |
The AC Trap: Voltage, Phase, and Power Factor
The math of G = 1 / R is absolute, but applying it to real-world AC power systems introduces severe pitfalls. Here is how to navigate the variables that dictate whether your conversion is useful or entirely misleading.
What assumption fixes the answer?
The conversion of resistance to conductance assumes a purely resistive DC circuit, or an AC circuit with exactly zero reactance (like a purely resistive heating element). If inductance or capacitance is present, you are no longer dealing with pure resistance (R); you are dealing with impedance (Z). The reciprocal of impedance is admittance (Y), not conductance.
How the answer shifts for 120V vs 230V vs 3-phase
The fundamental arithmetic of 1 / R does not shift based on voltage. However, the measurement context shifts dramatically. In a 120V single-phase branch circuit, you are typically measuring line-to-neutral resistance. In 230V single-phase or 480V 3-phase systems, you are measuring line-to-line. If you measure 10 Ω across two phases of a 3-phase induction motor, you are measuring the complex impedance of the windings, not the pure DC resistance. Treating that 10 Ω AC impedance as pure resistance and converting it to 0.1 S will result in drastically undersized conductors and breaker trips, because you have ignored the reactive component of the motor windings.
When the conversion is meaningless (e.g., pf unknown)
Converting resistance to conductance is practically meaningless in AC load calculations when the Power Factor (pf) is unknown. In AC theory, admittance (Y) is composed of conductance (G) and susceptance (B). If you measure an HVAC compressor and only extract the real resistance (R) without knowing the reactive component (X) or the power factor, calculating 1 / R only gives you the conductance of the resistive heating losses. It completely ignores the magnetic susceptance required to spin the motor. Without the power factor, you cannot calculate total admittance (Y = 1 / Z), making your conductance figure useless for sizing VFDs, contactors, or branch circuit breakers. For a rigorous breakdown of the admittance triangle, refer to Georgia State University's HyperPhysics module on complex impedance.
Frequently Asked Questions
How do I convert resistance to conductivity for a specific wire gauge?
You cannot directly convert the resistance of a specific wire gauge to material conductivity without knowing the wire's exact length and cross-sectional area. First, calculate the resistivity (ρ) using the formula ρ = (R × A) / L, where R is your measured resistance in ohms, A is the cross-sectional area in square meters, and L is the length in meters. Once you have resistivity (Ω·m), invert it (σ = 1 / ρ) to find the material conductivity in S/m. For standard 12 AWG copper wire (area ≈ 3.31 × 10⁻⁶ m²), a 100-meter spool should measure roughly 0.508 Ω at 20°C, yielding a resistivity of 1.68 × 10⁻⁸ Ω·m and a conductivity of 5.96 × 10⁷ S/m.
Is conductance the exact same thing as conductivity?
No. Conductance (G, measured in Siemens) is an extrinsic property of a specific, physical object—like a 10 Ω resistor or a 50-foot run of 14 AWG wire. If you cut the wire in half, its resistance drops, and its conductance doubles. Conductivity (σ, measured in S/m) is an intrinsic property of the material itself. The conductivity of copper remains 5.96 × 10⁷ S/m regardless of whether you have a microscopic trace on a PCB or a massive 500 kcmil feeder cable.
Why does my multimeter measure resistance but not conductance?
Digital multimeters (DMMs) like the Fluke 87V or Keysight U1232A measure resistance by sourcing a known, precise constant current through the probes and measuring the resulting voltage drop (using Ohm's Law, R = V / I). Because conductance is simply the mathematical inverse of that calculated resistance, meter manufacturers do not include a dedicated Siemens setting. It is considered redundant for field work. If you need to log conductance for a materials science test, you must export the resistance data to a spreadsheet or write a quick Python script to invert the values (G = 1 / R) post-measurement.
