The direct answer for single-phase AC or DC circuits is the standard approximate formula: VD = (2 × K × I × D) / CM. For a 120V, 15A load at a 100-foot one-way distance using 14 AWG copper wire, the voltage drop is 9.41V (7.8%). This massively exceeds the recommended 3% branch circuit limit, dictating an immediate upgrade to 10 AWG or 8 AWG depending on the exact terminal temperature rating. Below is the complete derivation, symbol definition, and step-by-step math to size your conductors correctly without guessing.

The Core Voltage Drop Calculation for Cable & Symbols

The formula used by electricians and engineers for single-phase systems is derived from Ohm’s Law (V = I × R). Because a single-phase circuit requires a current path out to the load and a return path back to the source, the one-way distance must be doubled. The resistance of the wire is determined by its material resistivity (K) and its cross-sectional area in Circular Mils (CM).

Symbol Definition Unit Typical Value / Source
VD Voltage Drop (the absolute voltage lost in the wire) Volts (V) Target: ≤ 3.6V for 120V circuits
2 Multiplier for single-phase out-and-back path Dimensionless Use 1.732 for 3-phase systems
K Direct-current constant (resistivity) of the conductor Ω·cmil/ft 12.9 for Copper, 21.2 for Aluminum (at 75°C)
I Load current (continuous or non-continuous) Amperes (A) Nameplate rating or calculated load
D One-way distance from source to load Feet (ft) Measured physical run length
CM Cross-sectional area of the wire in Circular Mils cmil 4,110 for 14 AWG; 16,510 for 8 AWG
When this formula applies and its assumptions: This calculation assumes steady-state DC resistance or 60Hz AC circuits using conductors smaller than 1/0 AWG, where AC reactance (skin effect and power factor) is negligible. It assumes the wire is operating at its rated temperature column (typically 75°C for modern NFPA 70 (NEC) terminations). If you are sizing cable for high-frequency AC, massive industrial feeders, or extreme ambient temperatures, you must use complex impedance (Z) tables instead.

Rearranged Forms: Solving for Wire Size, Distance, and Current

In practice, you rarely solve for VD. You usually know your allowable voltage drop and need to find the required wire size, or you know your wire size and need to find the maximum run distance. Here are the algebraically rearranged forms of the core equation:

  • Solving for CM (Required Wire Size):
    CM = (2 × K × I × D) / VD
    Use this to find the minimum Circular Mils required, then look up the next largest standard AWG size.
  • Solving for D (Maximum One-Way Distance):
    D = (CM × VD) / (2 × K × I)
    Use this to determine how far you can run a specific wire gauge before exceeding your drop limit.
  • Solving for I (Maximum Allowable Current):
    I = (CM × VD) / (2 × K × D)
    Use this to find the maximum load a specific wire run can handle while maintaining voltage regulation.

Worked Examples with Unit Tracking

Abstract formulas are useless without unit tracking. Let us run two real-world scenarios to demonstrate how the math dictates your physical wire purchase.

Problem 1: Evaluating an Existing 14 AWG Run

Scenario: A 120V nominal branch circuit powers a 15A space heater. The one-way cable run is 100 feet. The installed wire is 14 AWG Copper. What is the voltage drop, and does it pass the 3% NEC-style recommendation?

  1. Identify Variables: K = 12.9 (Copper at 75°C), I = 15A, D = 100 ft, CM = 4,110 (standard 14 AWG).
  2. Calculate Numerator: 2 × 12.9 × 15 × 100 = 38,700.
  3. Divide by Denominator (CM): 38,700 / 4,110 = 9.41V drop.
  4. Calculate Percentage: (9.41V / 120V) × 100 = 7.84%.

Verdict: A 7.84% drop is unacceptable for a branch circuit (target is ≤ 3%). The heater will receive only 110.59V, causing it to draw more current to compensate (if it has a switching power supply) or produce significantly less heat (if it is a pure resistive load). You must upsize the wire.

Problem 2: Sizing a New 240V Subpanel Feeder

Scenario: You are running a 240V, 30A dedicated circuit for a workshop welder. The one-way distance is 150 feet. You want to limit the voltage drop to exactly 3%. What wire size do you need?

  1. Identify Target VD: 3% of 240V = 0.03 × 240 = 7.2V maximum drop.
  2. Set up Rearranged Formula for CM: CM = (2 × K × I × D) / VD.
  3. Plug in Values: K = 12.9, I = 30A, D = 150 ft, VD = 7.2V.
  4. Calculate Numerator: 2 × 12.9 × 30 × 150 = 116,100.
  5. Divide by Target VD: 116,100 / 7.2 = 16,125 CM required.
  6. Select AWG: 10 AWG is 10,380 CM (too small). 8 AWG is 16,510 CM (passes).

Verdict: You must pull 8 AWG Copper. While 10 AWG is legally rated for 30A of ampacity under NEC 310.16, it fails the voltage drop calculation at this distance. Ampacity prevents the wire from melting; voltage drop ensures the equipment actually works.

Common Unit Mistakes That Break the Math

When your voltage drop calculations yield wildly unrealistic numbers, you have almost certainly committed one of these three unit errors:

The Mistake Why It Breaks the Math The Fix
Using Meters for Distance (D) The K factor (12.9) is calibrated for ohms per mil-foot. Plugging in meters shrinks your distance variable by ~3.28x, resulting in a falsely optimistic wire size. Always convert meters to feet (multiply meters by 3.28084) before plugging into the formula, or use a metric K factor and mm².
Confusing CM with mm² Circular Mils (CM) and square millimeters (mm²) are entirely different area measurements. 10 AWG is 10,380 CM, but only 5.26 mm². Swapping them destroys the denominator. Use a standard AWG-to-CM conversion chart. If your wire is metric, convert mm² to CM by multiplying by 1,973.5.
Forgetting the '2' Multiplier Single-phase current must return to the source. Forgetting the '2' calculates the drop for only half the physical circuit, underestimating the total drop by 50%. Always use '2' for single-phase/DC. Only drop the '2' (and use 1.732 instead) when calculating balanced 3-phase systems.
What a realistic answer magnitude looks like: On a standard 120V residential branch circuit, a realistic voltage drop is between 1.0V and 3.6V. If your calculation spits out 45V or 0.02V, your decimal placement or unit conversion is wrong. On a 240V feeder, expect 2.0V to 7.2V. Anything outside these magnitudes warrants a re-check of your inputs.

Decision Tree: Sizing Your Cable for a 3% Drop

Use this decision path to terminate your planning phase and pick a concrete wire size for a standard 120V, 20A receptacle circuit. This framework eliminates "it depends" guesswork and gets you to the hardware store.

Step Condition / Action Resulting Value
1. Define Load & Distance Load = 20A continuous. One-way distance = 80 feet. I = 20, D = 80
2. Set Target Voltage Drop Calculate 3% of 120V nominal source. VD = 3.6V max
3. Select Material & K Choose Copper for indoor residential branch wiring. K = 12.9
4. Calculate Required CM CM = (2 × 12.9 × 20 × 80) / 3.6 CM = 11,466
5. Check AWG Chart 12 AWG = 6,530 CM (Fail). 10 AWG = 10,380 CM (Fail). 8 AWG = 16,510 CM (Pass). Minimum 8 AWG required for 3%.
6. Verify Ampacity (NEC 310.16) Does 8 AWG THHN handle 20A at 75°C? Yes, rated for 50A. Ampacity check passes.
7. Final Concrete Pick Terminate decision path. Purchase wire. Buy 8 AWG THHN Copper (Black/White/Green).

While manufacturer calculators can automate this math, understanding the underlying algebra ensures you catch software errors, adapt to unusual K-factors for high-temperature environments, and confidently defend your wire sizing to an AHJ inspector. Always size for voltage drop first, then verify the chosen wire against the ampacity tables for thermal safety.