If you need the direct answer immediately: the standard DC-resistance approximation for the voltage drop 3 phase formula is VD = (√3 × K × I × L) / CM. Unlike single-phase calculations, the 3-phase geometry introduces the √3 (1.732) constant because you are measuring line-to-line voltage across a balanced wye or delta system, not line-to-neutral. Sizing wire purely on ampacity without checking voltage drop is a rookie mistake that leads to humming motors, dim lighting, and tripped breakers at the end of long feeder runs.
Below, we break down the exact formula, provide the reference tables you need to keep at the bench, and walk through two solved problems with strict unit tracking so you can see exactly how the math cancels out.
The Core 3-Phase Voltage Drop Formula & Symbol Definitions
The standard formula used for most commercial and industrial branch circuits (where wire reactance is negligible) is derived from Ohm’s Law (V = I × R), adapted for the physical properties of wire and 3-phase geometry.
VD = (√3 × K × I × L) / CM
| Symbol | Definition | Standard Unit | Notes & Typical Values |
|---|---|---|---|
| VD | Voltage Drop | Volts (V) | The actual lost voltage. Target < 3% of nominal system voltage. |
| √3 | 3-Phase Constant | Unitless | Approximately 1.732. Represents the phase angle shift in 3-phase systems. |
| K | Resistivity Constant | Ω·cmil/ft | Copper ≈ 12.9 (at 75°C); Aluminum ≈ 21.2 (at 75°C). |
| I | Current (Load) | Amperes (A) | Use the continuous load current (125% of rated if applicable). |
| L | One-Way Length | Feet (ft) | Distance from source to load. Do NOT double this for 3-phase. |
| CM | Circular Mils | cmil | Cross-sectional area of the wire. Found in NEC Chapter 9, Table 8. |
Reference Data: Wire Properties & Constants
You cannot solve the formula without accurate constants. The K factor changes with temperature. While pure copper at 20°C has a K of roughly 10.4, terminations in commercial panels are typically rated for 75°C. Wire heats up under load, increasing resistance. Therefore, we use the 75°C K values for realistic voltage drop calculations under load. Below is the essential data derived from NFPA 70 (NEC) Chapter 9, Table 8 and standard engineering wire gauge references.
| AWG / kcmil | Area (CM) | Copper K (75°C) | Aluminum K (75°C) | Max Ampacity (75°C, 3-wire) |
|---|---|---|---|---|
| 14 AWG | 4,110 | 12.9 | 21.2 | 20A |
| 10 AWG | 10,380 | 12.9 | 21.2 | 35A |
| 6 AWG | 26,240 | 12.9 | 21.2 | 65A |
| 2 AWG | 66,360 | 12.9 | 21.2 | 115A |
| 4/0 AWG | 211,600 | 12.9 | 21.2 | 230A |
| 250 kcmil | 250,000 | 12.9 | 21.2 | 255A |
Rearranged Forms for Sizing and Troubleshooting
On the jobsite, you rarely just want to find the voltage drop. Usually, you know the maximum allowable drop (e.g., 3% of 480V = 14.4V) and need to find the right wire size. Here are the algebraic rearrangements of the core formula.
- Solve for Wire Size (CM): Use this when designing a new feeder run.
CM = (√3 × K × I × L) / VD - Solve for Maximum Distance (L): Use this when placing a subpanel to ensure you don't exceed your drop budget.
L = (VD × CM) / (√3 × K × I) - Solve for Maximum Current (I): Use this when troubleshooting an existing circuit to see how much load you can add before violating the 3% rule.
I = (VD × CM) / (√3 × K × L)
Worked Examples with Unit Tracking
Let’s run two real-world scenarios. Tracking units is the only way to guarantee you haven't mixed up your constants.
Example 1: Calculating Voltage Drop on an Existing Feeder
Scenario: You have a 480V, 3-phase, 60A motor located 250 feet from the MCC. The feeder is 6 AWG Copper THHN. What is the voltage drop, and does it pass the 3% rule?
- Identify Variables: √3 = 1.732; K = 12.9 (Copper, 75°C); I = 60A; L = 250 ft; CM = 26,240 (for 6 AWG).
- Setup Equation:
VD = (1.732 [unitless] × 12.9 [Ω·cmil/ft] × 60 [A] × 250 [ft]) / 26,240 [cmil] - Cancel Units: The cmil in the K factor cancels the cmil in the denominator. The ft in the K factor cancels the ft in the length. We are left with Ω × A, which equals Volts.
- Calculate Numerator: 1.732 × 12.9 × 60 × 250 = 334,710
- Divide by CM: 334,710 / 26,240 = 12.75 V
- Check Percentage: (12.75V / 480V) × 100 = 2.65%
Verdict: 2.65% is under the 3% NEC recommendation. However, 6 AWG copper at 75°C is only rated for 65A. Since the motor is 60A, it is thermally safe, but barely. If the motor had a high inrush or was a continuous duty load requiring 125% sizing (75A), you would need to step up to 4 AWG regardless of voltage drop.
Example 2: Sizing Wire for a New 208V Subpanel
Scenario: You are feeding a 100A, 208V, 3-phase subpanel located 150 feet away. You must keep the voltage drop under 3%. What size Copper wire do you need?
- Calculate Max Allowable VD: 208V × 0.03 = 6.24 V.
- Identify Variables: √3 = 1.732; K = 12.9; I = 100A; L = 150 ft; VD = 6.24V.
- Setup Rearranged Equation:
CM = (1.732 × 12.9 × 100 × 150) / 6.24 - Calculate Numerator: 1.732 × 12.9 × 100 × 150 = 334,710
- Divide by VD: 334,710 / 6.24 = 53,639 cmil
- Select Wire: Looking at our reference table, 3 AWG is 52,620 cmil (too small). 2 AWG is 66,360 cmil.
Verdict: You must pull 2 AWG Copper. Always verify ampacity: 2 AWG at 75°C is rated for 115A, which safely covers the 100A breaker. (For deeper insights on how voltage dip affects downstream equipment, refer to Schneider Electric's power quality guidelines).
Assumptions, Edge Cases, and Fatal Unit Mistakes
The formula above is an approximation that works perfectly for 90% of commercial and residential 3-phase work. But if you step outside its boundaries, your math will lie to you.
When the Formula Applies (and When It Doesn't)
- Balanced Loads Only: This formula assumes the current on Phase A, B, and C is identical. If you are feeding a heavily unbalanced panel (e.g., mostly single-phase 120V lighting on Phase A), the neutral will carry current, and the voltage drop on the heavily loaded phase will be higher than this formula predicts.
- Negligible Reactance (Wires ≤ 1/0 AWG): For smaller wires, AC resistance (R) dominates. For large wires (2/0 AWG and larger) pulled through steel conduit, inductive reactance (X) becomes significant. The true AC formula is
VD = √3 × I × L × (R cosθ + X sinθ). If you are sizing 500 kcmil feeders in steel pipe, use a software tool or the NEC Chapter 9, Table 9 AC resistance values, not the simple DC K-factor.
Fatal Unit Mistakes That Break the Math
I've seen journeymen get voltage drop numbers that suggest a 150V drop on a 480V system. When the math yields a realistic magnitude (usually 2V to 15V depending on system voltage), you know you're in the ballpark. If you get 85V, check these traps:
- Mixing Meters and Feet: The K factor of 12.9 is strictly derived for feet. If your blueprint is in meters, you must convert the length to feet (multiply by 3.281) before plugging it into the formula, or use the metric resistivity formula (VD = √3 × I × L × ρ / A) where ρ is in Ω·m and A is in mm².
- Using mm² Instead of CM: Circular Mils (CM) and square millimeters (mm²) are not interchangeable. 10 AWG is 10,380 CM, but it is only 5.26 mm². Plugging 5.26 into the CM denominator will result in a calculated voltage drop that is roughly 2,000 times too high.
- Doubling the Length for 3-Phase: In single-phase (line-to-neutral) calculations, current travels out on the hot and back on the neutral, so you multiply the one-way distance by 2. In a balanced 3-phase system, the return path is handled by the phase shift of the other two conductors. Do not multiply L by 2. The √3 constant already accounts for the 3-phase geometry.
- Ignoring the 125% Continuous Load Rule: If your 3-phase load is a continuous duty motor or heater running for 3+ hours, the NEC requires you to size the wire for 125% of the load. If the motor draws 40A, you must use I = 50A in your voltage drop formula to reflect the actual thermal and resistive state of the wire under continuous operation.
Safety & Code Caveat: Voltage drop calculations are a matter of power quality and efficiency (NEC Informational Note 310.15(B)). They do not override the mandatory ampacity and overcurrent protection rules of NEC 310.16 and 240.4. A wire might have an excellent voltage drop profile over a short distance but still melt if it exceeds its temperature rating. Always calculate ampacity first, then check voltage drop, and use whichever yields the larger wire size.
