Voltage Drop Formula Explained: DC, AC & 3-Phase Calculations

The Basic Voltage Drop Equation

Understanding voltage drop begins with Ohm's Law, the fundamental relationship between voltage, current, and resistance. Georg Ohm discovered that the voltage across a conductor is directly proportional to the current flowing through it, with resistance as the proportionality constant.

When current flows through any conductor, it encounters resistance. This resistance causes some of the electrical energy to be converted to heat, resulting in a voltage drop along the length of the wire. The longer the wire and the higher the current, the greater the voltage drop.

In practical electrical installations, we need to account for the entire circuit path. Current flows from the source through the wire to the load, then returns through another wire back to the source. This means the total wire length is actually twice the one-way distance in most circuits.

Why Voltage Drop Matters

Voltage drop is crucial for several reasons. Equipment is designed to operate at specific voltages, and when the voltage at the load is too low, problems occur. Motors produce less torque and may overheat. Lights dim noticeably. Electronics may malfunction or fail to operate. Additionally, voltage drop represents wasted energy—the power lost to resistance appears as heat in the wires rather than useful work at the load.

The National Electrical Code recommends limiting voltage drop to 3% on branch circuits and 5% total (including feeders), ensuring equipment operates efficiently and safely. Understanding how to calculate voltage drop allows electricians and engineers to properly size conductors for their installations.

DC Voltage Drop Formula

For direct current (DC) circuits, the voltage drop formula is straightforward and based directly on Ohm's Law. DC current flows in one direction continuously, making the calculation relatively simple compared to AC circuits.

Where:

• VD = Voltage drop (volts)
• L = One-way length of the circuit (feet)
• I = Current (amperes)
• R = Wire resistance (ohms per 1000 feet)

Why the Factor of 2?

The factor of 2 accounts for the complete circuit path. Current must travel from the source to the load (one-way distance L) and then return from the load back to the source (another distance L). The total conductor length carrying current is therefore 2 × L. Some references show the formula as VD = (2 × L × I × R) / 1000, which is mathematically equivalent but uses different units for length.

Why Divide by 1000?

The division by 1000 is a unit conversion factor. Wire resistance values from NEC Chapter 9, Table 8 are given in ohms per 1000 feet. By dividing by 1000, we convert our length in feet to thousands of feet, making the units work out correctly. If you use resistance in ohms per foot instead, you would not divide by 1000.

DC Applications

The DC voltage drop formula applies to numerous applications including automotive electrical systems, battery banks, solar panel installations, marine and RV systems, and any other application using direct current. These systems are particularly sensitive to voltage drop because they typically operate at lower voltages (12V, 24V, 48V), where even small voltage drops represent a significant percentage of the total voltage.

Single-Phase AC Formula

For single-phase alternating current circuits, the formula includes an additional factor: power factor. AC circuits involve more complex interactions between voltage and current, and the power factor accounts for the phase relationship between them.

Where:

• VD = Voltage drop (volts)
• L = One-way length of the circuit (feet)
• I = Current (amperes)
• R = Wire resistance (ohms per 1000 feet)
• PF = Power factor (decimal, typically 0.85-1.0)

Understanding Power Factor

Power factor represents the ratio of real power (watts) to apparent power (volt-amperes) in an AC circuit. In a purely resistive circuit like an electric heater, voltage and current are perfectly in phase, and the power factor is 1.0. In circuits with motors or other inductive loads, current lags behind voltage, resulting in a power factor less than 1.0.

For most residential and commercial calculations, a power factor of 0.85 to 0.90 is commonly used for circuits feeding motors. For resistive loads like heaters and incandescent lights, use a power factor of 1.0. When the exact power factor is unknown, using 0.85 provides a conservative estimate that accounts for typical inductive loads.

Simplified AC Formula

For resistive loads or when a conservative general estimate is needed, many electricians simplify the formula by assuming a power factor of 1.0, making the AC formula identical to the DC formula. This approach always gives a voltage drop equal to or slightly higher than the actual value, providing a safety margin in the design.

Using Impedance Instead of Resistance

For greater accuracy in AC circuits, especially with larger conductors, you can use the impedance values from NEC Chapter 9, Table 9 instead of resistance values from Table 8. Table 9 accounts for the inductive reactance of conductors in conduit, which becomes significant for larger wire sizes and longer runs. The formula structure remains the same, but you substitute Z (impedance) for R (resistance).

Three-Phase AC Formula

Three-phase power systems are common in industrial and commercial applications because they can transmit more power with less conductor material than single-phase systems. The voltage drop formula for three-phase circuits differs from single-phase due to the geometry of how three-phase power works.

Where:

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

Why √3 Instead of 2?

The factor of √3 (approximately 1.732) appears in three-phase calculations because of the 120-degree phase relationship between the three conductors. In a balanced three-phase system, the currents in the three phases are offset by 120 degrees. When you analyze the voltage drop between any two phases (line-to-line voltage), the mathematical result involves the square root of 3.

To understand this intuitively, consider that in a three-phase system, current flows through three conductors instead of two. The return path for current is through the other two phases rather than a dedicated return conductor. This distributed return path means the effective circuit length is not simply 2 × L as in single-phase circuits, but rather √3 × L.

Line vs. Phase Voltage Drop

The three-phase formula calculates line-to-line voltage drop, which is the most commonly used value. If you need the phase voltage drop (line-to-neutral), divide the result by √3. For a 480V three-phase system, the line-to-line voltage is 480V and the line-to-neutral (phase) voltage is 277V.

Balanced vs. Unbalanced Loads

The three-phase formula assumes a balanced load, meaning equal current flows in all three phases. For significantly unbalanced loads, you may need to calculate voltage drop for each phase separately using the single-phase formula. In practice, most three-phase systems are designed to be reasonably balanced, and the standard formula provides adequate accuracy.

Understanding Each Variable

Length (L)

Length is the one-way distance from the power source to the load, measured in feet. This is the actual wire routing distance, not the straight-line distance. Include all bends, vertical runs, and routing around obstacles. When wire must run up and over obstructions or through conduit with bends, the actual length may be significantly longer than the direct distance.

For metric calculations, you can convert meters to feet by multiplying by 3.281, or use resistance values in ohms per kilometer and adjust the formula accordingly.

Current (I)

Current is the expected load current in amperes. For motor circuits, use the full-load current from the motor nameplate or NEC Table 430.250, not the circuit breaker rating. For general-purpose circuits, use the maximum expected load current. For continuous loads (operating for 3 hours or more), size the conductors for 125% of the continuous load per NEC requirements.

Remember that voltage drop occurs at the actual operating current, not the maximum circuit capacity. However, designing for maximum expected current ensures adequate voltage under all operating conditions.

Resistance (R)

Wire resistance depends on the conductor material (copper or aluminum), wire size (AWG or kcmil), and temperature. NEC Chapter 9, Table 8 provides DC resistance values for uncoated copper and aluminum conductors at 75°C. These values are in ohms per 1000 feet.

Copper has about 60% of the resistance of aluminum for the same wire size. This means copper conductors can be smaller than aluminum for the same current capacity and voltage drop. However, aluminum is lighter and less expensive, making it a practical choice for larger installations where the size difference is acceptable.

Power Factor (PF)

Power factor is a decimal value between 0 and 1 representing the phase relationship between voltage and current. Typical values include:

• 1.0: Resistive loads (heaters, incandescent lights)
• 0.90-0.95: LED lighting, computers, modern electronics
• 0.80-0.90: Motors at full load
• 0.60-0.80: Motors at partial load

When in doubt, use 0.85 for general calculations. This provides a reasonable estimate for mixed loads while remaining somewhat conservative.

Wire Resistance Values (NEC Table 8)

The following table shows DC resistance values from NEC Chapter 9, Table 8 for common wire sizes. These values are for uncoated copper and aluminum conductors at 75°C (167°F). Use these values with the voltage drop formulas presented above.

Note that aluminum wire has approximately 1.6 times the resistance of copper wire of the same size. To achieve equivalent performance to copper, aluminum conductors typically need to be two AWG sizes larger (for example, 2 AWG aluminum instead of 4 AWG copper).

Worked Examples

Example 1: DC Circuit (12V Solar Panel)

Problem: A 12V solar panel system needs to run 30 feet to a charge controller. The maximum current is 15A. Using 10 AWG copper wire, what is the voltage drop?

Given:

• L = 30 feet
• I = 15A
• R = 1.24 Ω/1000ft (10 AWG copper)

Solution:

Analysis: The voltage drop is 1.12V, which is 9.3% of the 12V system voltage. This exceeds the recommended 3% maximum. A larger wire size (8 AWG or 6 AWG) would be needed to reduce voltage drop to acceptable levels.

Example 2: Single-Phase AC (120V Branch Circuit)

Problem: A 120V, 20A circuit runs 75 feet to kitchen outlets. Using 12 AWG copper wire and assuming a power factor of 1.0 for resistive loads, what is the voltage drop?

Given:

• L = 75 feet
• I = 20A
• R = 1.98 Ω/1000ft (12 AWG copper)
• PF = 1.0

Solution:

Analysis: The voltage drop is 5.94V, which is 4.95% of 120V. This just exceeds the 3% recommendation for branch circuits. Using 10 AWG wire would reduce the drop to approximately 3.72V (3.1%), which is borderline. For a cleaner design, consider 10 AWG or reducing the circuit length.

Example 3: Three-Phase (480V Motor Circuit)

Problem: A 480V, three-phase motor draws 65A at full load. The motor is 200 feet from the panel. Using 4 AWG copper wire with a power factor of 0.85, what is the voltage drop?

Given:

• L = 200 feet
• I = 65A
• R = 0.308 Ω/1000ft (4 AWG copper)
• PF = 0.85

Solution:

Analysis: The voltage drop is 5.90V, which is only 1.23% of 480V. This is well within the 3% recommendation. The higher system voltage makes it much easier to maintain low percentage voltage drop.

Formula Comparison

Common Formula Mistakes

Understanding common errors helps avoid them in your own calculations. Here are the most frequent mistakes made when calculating voltage drop.

Mistake 1: Forgetting the Factor of 2

The most common error is forgetting to multiply by 2 in DC and single-phase AC calculations. Remember that current flows to the load AND back to the source, so the total conductor length is twice the one-way distance. Forgetting this factor will underestimate the voltage drop by half.

Mistake 2: Using Wrong Resistance Units

NEC Table 8 gives resistance in ohms per 1000 feet. If you forget to divide by 1000 in the formula, your result will be 1000 times too large. Conversely, if you use resistance in ohms per foot but still divide by 1000, your result will be 1000 times too small. Always verify your resistance values and units match the formula you're using.

Mistake 3: Mixing Metric and Imperial Units

Ensure all length measurements are in the same units as your resistance values. If your length is in meters but your resistance is in ohms per 1000 feet, you must convert. Multiply meters by 3.281 to get feet, or use resistance values in ohms per kilometer with lengths in meters.

Mistake 4: Using Single-Phase Formula for Three-Phase

Three-phase circuits use √3 (1.732) instead of 2 as the multiplier. Using the factor of 2 for three-phase calculations will overestimate voltage drop by about 15%. While this is conservative and won't cause safety issues, it may lead to unnecessarily large wire sizes and increased costs.

Mistake 5: Ignoring Power Factor

For AC circuits with significant motor loads, ignoring power factor can underestimate the actual voltage drop during operation. While using PF = 1.0 gives a quick conservative estimate for resistive loads, inductive loads with lower power factors may actually experience the calculated drop.

Mistake 6: Using Breaker Size Instead of Load Current

The voltage drop calculation should use the actual or expected load current, not the circuit breaker rating. A 20A breaker doesn't mean the circuit always carries 20A. Using actual load current gives more accurate results for sizing decisions. However, for worst-case analysis or future expansion, using the maximum circuit capacity is appropriate.

Mistake 7: Not Accounting for Temperature

NEC Table 8 resistance values are at 75°C. In very hot or cold environments, resistance changes. Resistance increases with temperature—copper resistance increases about 0.4% per degree Celsius above the reference temperature. For critical applications in extreme temperatures, adjust the resistance values accordingly.

Frequently Asked Questions

Sources

• NFPA 70: National Electrical Code (NEC) - Chapter 9, Tables 8 and 9
• IEEE 141 (Red Book): Recommended Practice for Electric Power Distribution
• Southwire Voltage Drop Calculator - Industry Reference Tool

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