3-Phase Voltage Drop Calculation: Formula, Examples & Motor Sizing

Three-Phase Power Fundamentals

Three-phase power is the standard for industrial and commercial electrical distribution because it's more efficient than single-phase for transmitting large amounts of power. Understanding the basics of three-phase systems helps explain why voltage drop calculations differ from single-phase circuits.

What Makes It Three-Phase?

A three-phase system consists of three conductors (phases), each carrying an AC current that's offset from the others by 120 electrical degrees. This means at any instant, the currents in the three phases are at different points in their cycles—when one phase is at its peak positive value, the other two are at lesser values. This arrangement provides several advantages over single-phase power.

The power delivered by a three-phase system is constant rather than pulsating as in single-phase. This makes three-phase ideal for motors, which run smoother and more efficiently on three-phase power. Additionally, three-phase systems can deliver the same power with less conductor material than single-phase, reducing installation costs for high-power applications.

Common Three-Phase Voltages

In North America, common three-phase voltages include:

• 208Y/120V: Common in commercial buildings; provides 208V three-phase and 120V single-phase
• 480Y/277V: Standard industrial voltage; provides 480V three-phase and 277V single-phase (for lighting)
• 240V Delta: Older industrial systems; three-phase only, no neutral
• 600V: Used in some industrial applications, particularly in Canada

Wye vs. Delta Configurations

Three-phase systems can be configured as either wye (Y) or delta (Δ). In a wye system, each phase connects from a neutral point to a line conductor, providing both line-to-line and line-to-neutral voltage. In a delta system, phases connect in a triangle pattern with no neutral point. Both configurations have the same line-to-line voltage drop characteristics for balanced loads.

The wye configuration is more common in modern installations because it provides a neutral connection for single-phase loads. The voltage drop formula works the same for both configurations when calculating line-to-line voltage drop, which is the standard measure for three-phase systems.

Why the √3 Factor is Used

One of the most common questions about three-phase voltage drop is why we use √3 (approximately 1.732) instead of the factor of 2 used in single-phase calculations. The answer lies in the geometry of how current flows in a three-phase system.

The Single-Phase Comparison

In a single-phase circuit, current flows from the source through one conductor to the load, then returns through a second conductor (neutral or the other "hot" leg in 240V circuits). The total wire length carrying current is twice the one-way distance, hence the factor of 2 in the single-phase formula.

The Three-Phase Difference

In a balanced three-phase system, the situation is more complex. Current flows through each of the three phase conductors, but there's no dedicated return path—the return for each phase occurs through the other two phases. Due to the 120-degree phase offset, the vector sum of currents in any two phases provides the return path for the third.

When you calculate the voltage drop between any two phases (line-to-line voltage), the mathematical analysis involves adding voltage drops that are offset by 120 degrees. The vector addition of these components results in the √3 factor. This is a fundamental property of three-phase geometry.

Mathematical Derivation

For those interested in the mathematics: if we consider the voltage drop in one phase conductor as V, the voltage drop between two phases (line-to-line) involves both conductors. Because the currents and voltages are 120 degrees apart, the resulting line-to-line voltage drop is V × √3, not 2V. This can be verified using phasor analysis or by examining the geometry of an equilateral triangle formed by the three phase vectors.

Practical Implications

The √3 factor (1.732) is smaller than 2, which means three-phase systems have inherently lower voltage drop than single-phase systems carrying the same current over the same distance. This is one reason three-phase is preferred for high-power, long-distance applications—you get the same power delivered with less voltage drop (and less wasted energy) than with single-phase.

Three-Phase Voltage Drop Formula

The standard formula for calculating voltage drop in three-phase circuits is:

Where:

• VD = Voltage drop (volts, line-to-line)
• √3 = Square root of 3 (1.732)
• L = One-way length of the circuit (feet)
• I = Line current (amperes)
• R = Wire resistance (ohms per 1000 feet) from NEC Table 8
• PF = Power factor (decimal, typically 0.85 for motors)

Using the Formula

This formula calculates the line-to-line voltage drop, which is the standard measurement for three-phase systems. To find the voltage at the load, subtract the voltage drop from the source voltage. For example, if a 480V system has 12V drop, the load receives 480V - 12V = 468V.

Voltage Drop Percentage

To convert to percentage:

The NEC recommends maximum 3% voltage drop for branch circuits and 5% total for feeders plus branch circuits. For motor circuits, staying within 3% helps ensure proper starting torque and running efficiency.

Alternative Formula Using Impedance

For greater accuracy, particularly with larger conductors where inductive reactance is significant, use impedance (Z) from NEC Table 9 instead of resistance:

Table 9 provides effective impedance values that account for both resistance and reactance of conductors in various conduit types. This is particularly important for conductors larger than 1/0 AWG and for long runs in steel conduit where inductive effects are more pronounced.

Balanced vs. Unbalanced Loads

The three-phase voltage drop formula assumes a balanced load—equal current in all three phases. Real-world systems often have some degree of imbalance, which affects voltage drop calculations.

Balanced Load Conditions

A perfectly balanced three-phase load draws equal current in all three phases at identical power factors. Three-phase motors are inherently balanced loads, making the standard formula ideal for motor circuit calculations. In a balanced system, the neutral current is zero (in wye systems), and each phase experiences identical voltage drop.

Most industrial three-phase loads—motors, large HVAC equipment, and three-phase heating elements—are balanced. When calculating voltage drop for these loads, the standard three-phase formula gives accurate results.

Unbalanced Load Conditions

Unbalanced loads have different currents in each phase. This commonly occurs when single-phase loads are connected to a three-phase system, or when one phase of a motor fails. In unbalanced conditions, each phase has different voltage drop, and neutral current flows in wye systems.

For significantly unbalanced loads, calculate voltage drop for each phase separately using the single-phase formula with the actual current in that phase. The phase with the highest current will have the greatest voltage drop. This approach gives a more accurate picture of system performance under unbalanced conditions.

Neutral Conductor Sizing

In four-wire wye systems feeding single-phase loads, the neutral conductor carries the imbalance current. For systems with significant single-phase loads, the neutral must be sized for the expected imbalance. Heavily loaded neutrals can experience their own voltage drop, affecting the voltage available to single-phase loads.

Harmonic-producing loads like computers and LED lighting can cause neutral currents to exceed phase currents due to triplen harmonics (3rd, 9th, 15th, etc.). In these cases, the neutral may need to be larger than the phase conductors, or a separate neutral may be required for harmonic-heavy circuits.

Industrial Motor Calculations

Electric motors are the most common three-phase loads and have specific characteristics that affect voltage drop calculations. Proper wire sizing ensures motors start reliably and run efficiently.

Motor Full-Load Amperes

The National Electrical Code provides tables of motor full-load amperes (FLA) in Article 430. These values should be used for voltage drop calculations rather than nameplate values, which may vary. The following table shows FLA for common three-phase motor sizes at 460V (used for 480V systems):

Starting Current Considerations

Motors draw 6-8 times their full-load current during starting. This inrush current causes temporary voltage drop that can exceed the 3% guideline. For most motors with normal starting duty, this temporary drop is acceptable as long as running voltage drop is within limits. However, for motors that start frequently or under heavy load, consider the starting current when sizing conductors.

Excessive voltage drop during starting can cause motors to fail to start, especially under load. If voltage drops below about 70% of rated voltage, a motor may stall or not develop enough torque to accelerate. For installations with long wire runs to motor loads, verify that starting voltage drop doesn't cause operational problems.

Power Factor for Motors

Motor power factor varies with load. At full load, most induction motors operate at 0.80-0.90 power factor. At light loads, power factor drops significantly, sometimes to 0.50 or below. For voltage drop calculations, using 0.85 is a reasonable average for motors running at typical loads.

Worked Examples

Example 1: 480V Motor Circuit

Problem: A 25 HP motor operates at 480V three-phase. The motor is 150 feet from the panel. Using 8 AWG copper wire and a power factor of 0.85, calculate the voltage drop.

Given:

• L = 150 feet
• I = 34A (from motor FLA table)
• R = 0.778 Ω/1000ft (8 AWG copper)
• PF = 0.85

Solution:

Analysis: Voltage drop is 5.84V, which is 1.22% of 480V. This is well within the 3% recommendation. The motor will receive 474.16V at full load.

Example 2: 208V Commercial System

Problem: A three-phase air handling unit draws 45A at 208V. The run from the panel to the unit is 100 feet. What wire size is needed for 3% maximum voltage drop?

Given:

• L = 100 feet
• I = 45A
• Maximum VD = 3% of 208V = 6.24V
• PF = 0.90

Solution: Rearrange the formula to solve for maximum resistance:

Analysis: Looking at the wire resistance table, 8 AWG copper has R = 0.778 Ω/1000ft, which is less than the maximum of 0.890. Therefore, 8 AWG copper is adequate. 6 AWG (R = 0.491) would provide even more margin.

Single-Phase vs. Three-Phase Comparison

To illustrate the advantage of three-phase, compare voltage drop for the same power delivered by single-phase vs. three-phase over 200 feet using 6 AWG copper:

This comparison shows why industrial facilities use 480V three-phase—the same power can be delivered with dramatically lower voltage drop, allowing smaller conductors for long runs.

Three-Phase Wire Sizing Table

The following table provides recommended wire sizes for three-phase 480V circuits based on current and distance, targeting 3% maximum voltage drop with 0.85 power factor:

These values are for copper conductors. For aluminum, use approximately two sizes larger. Always verify that the selected wire size also meets ampacity requirements from NEC Table 310.16. Use our voltage drop calculator for precise calculations for your specific conditions.

Frequently Asked Questions

Sources

• NFPA 70: National Electrical Code - Article 430 (Motors) and Chapter 9
• IEEE 141 (Red Book): Electric Power Distribution for Industrial Plants
• NEMA MG 1: Motors and Generators

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