The standard approximate 3 phase voltage drop formula used on the jobsite and in the NEC is VD = (√3 × K × I × L) / CM. This equation calculates the line-to-line voltage drop in a balanced, steady-state 3-phase AC circuit. While single-phase calculations use a multiplier of 2, the 120-degree phase shift in a 3-phase system reduces the effective multiplier to the square root of 3 (approximately 1.732). Understanding how to manipulate this formula is the difference between a motor that runs cool and one that burns out from undervoltage.
The Core Formula and Symbol Definitions
Before punching numbers into a calculator, you need to know exactly what each variable represents and the assumptions baked into the math. This formula assumes a balanced load, a steady-state current, and neglects AC reactance (which is acceptable for wires smaller than 1/0 AWG, but requires the exact impedance formula for larger feeders).
| Symbol | Definition | Standard Jobsite Value / Unit |
|---|---|---|
| VD | Voltage Drop (line-to-line) | Volts (V) |
| √3 | 3-phase phase-shift multiplier | 1.732 (dimensionless) |
| K | Conductor resistivity constant | 12.9 Ω·cmil/ft (Copper at 75°C) 21.2 Ω·cmil/ft (Aluminum at 75°C) |
| I | Load current | Amperes (A) |
| L | One-way length of the circuit | Feet (ft) |
| CM | Cross-sectional area in Circular Mils | cmil (Lookup via NEC Chapter 9, Table 8) |
Rearranged Forms: Solving for Wire Size, Distance, and Current
You rarely use the formula just to find the voltage drop; usually, you know your acceptable drop limit and need to find the right wire or maximum distance. Here are the algebraic rearrangements you will actually use in the field:
- To find required Wire Size (CM):
CM = (√3 × K × I × L) / VD - To find Maximum One-Way Distance (L):
L = (VD × CM) / (√3 × K × I) - To find Maximum Current (I):
I = (VD × CM) / (√3 × K × L)
Note: Once you calculate the required CM, you must round up to the next standard AWG size found in NEC Chapter 9, Table 8, and then verify that the chosen wire's ampacity (NEC Table 310.16) exceeds your load current.
Worked Problem 1: Sizing a Feeder for a 480V CNC Machine
Scenario: You are running a new 480V, 3-phase feeder to a CNC mill. The machine draws a continuous 60A. The one-way distance from the panel to the disconnect is 250 feet. You are using 75°C rated copper THHN in conduit. What is the voltage drop if you use 4 AWG wire, and is it acceptable?
Step 1: Gather and track the variables.
- √3 = 1.732
- K = 12.9 Ω·cmil/ft (Copper)
- I = 60 A
- L = 250 ft
- CM = 41,740 cmil (4 AWG copper, per NEC Table 8)
Step 2: Plug into the formula with units.
VD = (1.732 × 12.9 Ω·cmil/ft × 60 A × 250 ft) / 41,740 cmil
Step 3: Calculate the numerator (canceling 'ft' and 'cmil' leaves Volts).
Numerator = 334,980 V·cmil
Step 4: Divide by the denominator.
VD = 334,980 / 41,740 = 8.02 V
Step 5: Calculate the percentage.
%VD = (8.02 V / 480 V) × 100 = 1.67%
Verdict: 1.67% is well under the 3% NEC recommendation for feeders. 4 AWG is electrically sound for voltage drop, provided it meets the 75A minimum ampacity requirement for a 60A continuous load (which 4 AWG THHN at 75°C, rated 85A, easily satisfies).
Worked Problem 2: Finding Maximum Run Length for a 208V HVAC Unit
Scenario: A rooftop 208V, 3-phase RTU (Roof Top Unit) draws 45A. You have a spool of 6 AWG copper wire. What is the absolute maximum one-way distance you can run this wire while keeping the voltage drop at or below 3%?
Step 1: Define the limit and variables.
- Max VD = 3% of 208V = 6.24 V
- √3 = 1.732
- K = 12.9 Ω·cmil/ft
- I = 45 A
- CM = 26,240 cmil (6 AWG copper)
Step 2: Use the rearranged formula for Length (L).
L = (VD × CM) / (√3 × K × I)
Step 3: Substitute and solve.
L = (6.24 V × 26,240 cmil) / (1.732 × 12.9 Ω·cmil/ft × 45 A)
L = 163,737.6 / 1,002.8
L = 163.2 feet
Verdict: You can run the 6 AWG wire up to 163 feet. If the roof penetration is 180 feet away, you must step up to 4 AWG to avoid starving the compressor contactor of voltage.
Real-World Scenario: The 60HP Motor Starting Dip Disaster
Formulas look great on paper, but the bench and the jobsite have a way of exposing hidden assumptions. Here is a classic failure mode involving the 3 phase voltage drop formula.
The Setup: An industrial shop installed a 60HP, 460V, 3-phase air compressor. The Full Load Amps (FLA) on the nameplate was 77A. The run from the MCC (Motor Control Center) to the starter was 400 feet. The electrician sized the wire using the standard formula at FLA.
The Numbers: Using 2 AWG copper (CM = 66,360), the steady-state voltage drop calculation looked perfect:
VD = (1.732 × 12.9 × 77A × 400ft) / 66,360 = 10.3V (2.2% drop).
The Outcome: On the first start-up, the compressor motor hummed violently, stalled, and tripped the 150A breaker instantly. The voltage at the starter terminals sagged to under 380V during the start sequence.
What Went Wrong: The standard formula assumes steady-state FLA. It completely ignores Locked Rotor Amperage (LRA). A standard NEMA Design B motor draws roughly 6 times its FLA when starting across-the-line.
Let's run the formula using the LRA (77A × 6 = 462A):
Starting VD = (1.732 × 12.9 × 462A × 400ft) / 66,360 = 62.2V.
A 62.2V drop on a 460V system is a 13.5% voltage sag. Because motor starting torque is proportional to the square of the voltage, a 13.5% drop results in a roughly 25% loss of starting torque. The compressor couldn't break static friction, stalled, and drew locked-rotor current until the thermal mag-trip opened. Always calculate voltage drop at LRA for large motors to ensure it stays below 10-15% during startup, as noted in motor starting torque guidelines.
Unit Mistakes That Break the Math (And How to Avoid Them)
When your calculator spits out a voltage drop of 4,000V on a 208V system, you didn't break physics; you broke your units. Here are the three most common traps:
- Using AWG instead of Circular Mils (CM): The formula requires the cross-sectional area in CM, not the AWG gauge number. Plugging '4' into the CM slot instead of '41,740' will yield a mathematically absurd result. Always reference NEC Chapter 9, Table 8.
- Confusing Line-to-Line with Line-to-Neutral: The √3 multiplier inherently calculates the line-to-line drop. If you are analyzing a 3-phase wye system and need the line-to-neutral drop (for 120V/277V single-phase loads derived from it), you drop the √3 and use a multiplier of 2, or simply divide your 3-phase line-to-line result by √3.
- Using Meters instead of Feet with the standard K constant: The K value of 12.9 is specifically derived for feet. If your blueprint is in meters, you must either convert the distance to feet (multiply meters by 3.281) or use the metric resistivity formula where ρ (rho) is used in Ω·mm²/m. Mixing metric distance with imperial K constants is the #1 cause of bench-test failures.
By tracking your units through every step of the calculation and respecting the physical limits of motor starting currents, the 3 phase voltage drop formula transitions from a textbook exercise into a reliable tool for designing robust, code-compliant electrical systems.
