Wire Size Calculator Math: The Voltage Drop Formula & Worked Examples

Every online wire size calculator relies on one foundational algebraic relationship derived from Ohm's Law: the Circular Mil voltage drop formula. For single-phase AC or DC circuits, the required wire size in Circular Mils (CM) is calculated as: CM = (2 × K × I × L) / Vd. If you are sizing a 240V, 40A EV charger on a 100-foot run using copper wire (K=12.9) with a 3% maximum voltage drop (7.2V), the math demands 14,333 CM—which lands you squarely on 8 AWG wire. Understanding this formula prevents you from blindly trusting app outputs that might ignore thermal limits or assume incorrect operating temperatures.

The Core Voltage Drop Formula and Symbol Definitions

The formula calculates the minimum cross-sectional area required to keep voltage drop within acceptable limits. It applies strictly to single-phase AC (with high power factor) or DC circuits. It assumes a balanced load and ignores AC reactance (skin effect), which is a safe assumption for wires smaller than 1/0 AWG. For three-phase systems, the multiplier '2' is replaced by the square root of 3 (1.732).

Standard Wire Gauge Data: Circular Mils and Resistivity

To use the formula, you need accurate baseline data. The 'K' value changes with temperature. While cold copper (20°C) has a K of roughly 10.8, wires heat up under load. Professional wire size calculators use the 75°C column values because that reflects real-world operating temperatures inside conduit and panels. Below is the reference data you need to translate your CM answer into a physical wire size.

Rearranged Forms and Unit Mistakes That Break the Math

On the jobsite, you aren't always solving for wire size. Sometimes you need to know how far you can run a specific wire, or what the actual voltage drop will be on an existing circuit. Here are the rearranged forms solving for each variable:

• Solve for Voltage Drop (Vd): Vd = (2 × K × I × L) / CM
• Solve for Length (L): L = (Vd × CM) / (2 × K × I)
• Solve for Current (I): I = (Vd × CM) / (2 × K × L)
• Solve for Resistivity (K): K = (Vd × CM) / (2 × I × L)

Unit Mistakes That Will Ruin Your Calculation

I have seen apprentices fry 10 AWG wire on a 40A load because they trusted a calculator but fed it the wrong units. Watch out for these specific traps:

1. Using Meters Instead of Feet: The K constant (12.9 for Cu) is calibrated for feet. If you input 30 meters instead of 98.4 feet, your calculated CM will be roughly one-third of what it should be, resulting in a dangerously undersized wire.
2. Inputting Percentage Instead of Volts: Vd must be in absolute Volts. If you want a 3% drop on a 120V circuit, Vd is 3.6V. If you type '3' into the Vd variable, your math will be off by a factor of 1.2.
3. Using Breaker Size for 'I': 'I' is the actual continuous load, not the breaker rating. A 20A breaker protecting a 16A continuous load requires you to calculate using 16A (or 20A if applying the 125% continuous load rule upfront), not 20A blindly.

Worked Examples: Sizing Real-World Branch Circuits

Let's run through two common scenarios with strict unit tracking to show how the math dictates the physical wire you pull from the spool.

Problem 1: 120V Shed Subpanel Feeder

Scenario: You are running a 120V, 20A multi-wire branch circuit to a backyard shed. The one-way trench distance is 150 feet. You are using copper THHN and want to limit voltage drop to 3%.

1. Define Variables:
   • K = 12.9 Ω·cmil/ft (Copper at 75°C)
   • I = 20 A
   • L = 150 ft
   • Vd = 120V × 0.03 = 3.6 V
2. Substitute into Formula:
   CM = (2 × 12.9 Ω·cmil/ft × 20 A × 150 ft) / 3.6 V
3. Intermediate Math (Numerator):
   2 × 12.9 × 20 × 150 = 77,400 Ω·cmil·A (which simplifies to Volt·cmil)
4. Final Division:
   CM = 77,400 / 3.6 = 21,500 cmil
5. Table Lookup & Ampacity Check:
   Looking at our data table, 10 AWG is 10,380 cmil (too small). 8 AWG is 16,510 cmil (too small). 6 AWG is 26,240 cmil. You must pull 6 AWG copper to maintain a 3% drop. Since 6 AWG is rated for 65A at 75°C, it easily handles the 20A load thermally.

Problem 2: 240V EV Charger Hardwire

Scenario: You are wiring a Level 2 EV charger rated at 40A continuous (requiring a 50A circuit per NEC 210.20(A)). The panel is 200 feet away. You are using copper wire and the manufacturer mandates a maximum 3% voltage drop.

1. Define Variables:
   • K = 12.9 Ω·cmil/ft
   • I = 40 A (Use the actual continuous load, not the 50A breaker)
   • L = 200 ft
   • Vd = 240V × 0.03 = 7.2 V
2. Substitute into Formula:
   CM = (2 × 12.9 Ω·cmil/ft × 40 A × 200 ft) / 7.2 V
3. Intermediate Math (Numerator):
   2 × 12.9 × 40 × 200 = 206,400
4. Final Division:
   CM = 206,400 / 7.2 = 28,666 cmil
5. Table Lookup & Ampacity Check:
   6 AWG is 26,240 cmil (just under our requirement, meaning it would yield a 3.3% drop). 4 AWG is 41,740 cmil. You must select 4 AWG copper. Furthermore, a 40A continuous load requires a 50A breaker. 4 AWG copper is rated for 85A at 75°C, satisfying both the voltage drop math and the thermal ampacity rules.

When to Stop Calculating and Consult the AHJ

The circular mil formula is the engine inside tools like the Southwire Voltage Drop Calculator, but it is not the final authority on your jobsite. The NEC treats voltage drop as an informational note for branch circuits (NEC 210.19(A)(Informational Note No. 4)) recommending a 3% limit, but it becomes a strict, enforceable requirement for sensitive equipment or when specifically mandated by the manufacturer's installation instructions (NEC 110.3(B)).

Furthermore, this basic algebraic formula breaks down when you exceed 1/0 AWG. At larger wire diameters, AC skin effect and proximity effect increase the effective impedance (Z) beyond the basic DC resistance (K). For heavy commercial feeders, you must abandon the simple CM formula and use the AC impedance tables found in NEC Chapter 9, Table 9, factoring in the specific conduit material (PVC vs. steel) and power factor of the load. For standard residential and light commercial DIY work, however, mastering the CM formula ensures you will never undersize a long run again.