The AC Voltage Drop Equation: Derivation, Worked Examples, and Wire Sizing

The AC voltage drop equation calculates the loss of electrical potential across a conductor due to impedance. Unlike DC circuits where only resistance matters, AC circuits must account for both resistance and reactance, heavily influenced by the load's power factor and the magnetic properties of the conduit. For single-phase AC systems, the direct answer for calculating voltage drop is the linear projection formula derived from the phasor diagram: V_drop = 2 × I × (L / 1000) × (R_ac cosθ + X_L sinθ). This formula is the industry standard for NEC-compliant wire sizing, balancing mathematical accuracy with practical field application.

The Exact AC Voltage Drop Equation and Symbol Definitions

When sizing wire for alternating current, we use the effective impedance formula. The factor of '2' accounts for the out-and-return path of a single-phase circuit (Line and Neutral, or Line 1 and Line 2). Below is the complete spec-sheet definition for every symbol in the equation.

Symbol	Definition	Standard Unit	Source / Notes
V_drop	Voltage Drop	Volts (V)	The total potential lost across the conductors.
2	Multiplier	Dimensionless	Accounts for the out-and-return path (single-phase).
I	Load Current	Amperes (A)	The continuous or maximum expected load current.
L	One-Way Length	Feet (ft)	Distance from source to load, not total wire length.
1000	Divisor	Dimensionless	Converts table values (per 1,000 ft) to per-foot values.
R_ac	AC Resistance	Ω / 1,000 ft	From NEC Chapter 9, Table 9 (includes skin effect).
X_L	AC Reactance	Ω / 1,000 ft	From NEC Chapter 9, Table 9 (depends on conduit material).
cosθ	Power Factor	Dimensionless (0-1)	Ratio of real power to apparent power.
sinθ	Reactive Factor	Dimensionless (0-1)	Calculated as √(1 - cos²θ).

Application Boundaries, Assumptions, and Fatal Unit Mistakes

This formula is not a universal law; it is an engineered approximation that applies under specific conditions. Understanding these boundaries prevents catastrophic sizing errors.

When the Formula Applies (And Its Assumptions)

Steady-State Sinusoidal AC: It assumes a clean 60Hz (or 50Hz) sine wave. It does not apply to variable frequency drives (VFDs) outputting chopped PWM waveforms, where high-frequency harmonics drastically increase skin effect and effective resistance.
Single-Phase Systems: The '2' multiplier is strictly for 1-phase, 2-wire or 1-phase, 3-wire (line-to-line) systems. For balanced 3-phase systems, the multiplier changes to √3 (approx 1.732).
Negligible Phase Angle Shift: This is the 'approximate' AC voltage drop (the linear projection of the phasor). It assumes the angle between the sending and receiving voltage vectors is near zero, which is true for all standard residential and commercial wiring runs under 500 feet.

Unit Mistakes That Break the Math

⚠ Callout: The Three Fatal Errors

The /1000 Trap: NEC Table 9 lists R_ac and X_L per 1,000 feet. If you plug in 1.98 directly instead of 0.00198, your V_drop result will be 1,000 times too high.
Table 8 vs. Table 9: Using DC resistance (NEC Table 8) for AC circuits ignores skin effect and proximity effect. For 12 AWG, the difference is minor, but for 500 kcmil, AC resistance can be 20% higher than DC.
Metric Mixing: This specific arrangement relies on feet and Ω/kft. If your length is in meters, you must convert L to feet, or source R_ac and X_L in Ω/km and drop the 1000 divisor.

What a Realistic Answer Magnitude Looks Like

For a properly sized 120V branch circuit, a realistic V_drop is between 1.5V and 3.5V (1.25% to 2.9%). If your calculation yields 0.02V, you likely forgot to multiply by the current. If it yields 45V, you forgot the /1000 divisor or selected the wrong wire column. Per NFPA 70 (NEC) Informational Note 210.19(A), 3% is the target maximum for branch circuits, and 5% total for feeder + branch combined.

Worked Example 1: 120V Branch Circuit (Motor Load)

Scenario: You are wiring a 120V, 15A continuous-duty compressor motor. The one-way run length is 80 feet. The wire is 12 AWG Copper, installed in PVC conduit. The motor nameplate lists a Power Factor (cosθ) of 0.80.

Step 1: Gather NEC Table 9 Constants
From NEC Chapter 9, Table 9 (Copper, PVC conduit, 60Hz):
• R_ac for 12 AWG = 1.98 Ω/kft
• X_L for 12 AWG = 0.068 Ω/kft

Step 2: Calculate Reactive Factor (sinθ)
• cosθ = 0.80
• sinθ = √(1 - 0.80²) = √(1 - 0.64) = √0.36 = 0.60

Step 3: Calculate Effective Impedance (Z_eff)
• Z_eff = (R_ac × cosθ) + (X_L × sinθ)
• Z_eff = (1.98 × 0.80) + (0.068 × 0.60)
• Z_eff = 1.584 + 0.0408 = 1.6248 Ω/kft

Step 4: Execute the Master Equation
• V_drop = 2 × I × (L / 1000) × Z_eff
• V_drop = 2 × 15A × (80ft / 1000) × 1.6248 Ω/kft
• V_drop = 30 × 0.08 × 1.6248
• V_drop = 2.4 × 1.6248 = 3.899 Volts

Step 5: Calculate Percentage
• %VD = (3.899V / 120V) × 100 = 3.25%
Result: This exceeds the 3% NEC recommendation for a branch circuit. 12 AWG is insufficient.

Worked Example 2: 240V Subpanel Feeder (Mixed Load)

Scenario: A 240V, 40A subpanel feeder supplies a mixed load (lighting and receptacles). The one-way run is 150 feet. The wire is 6 AWG Copper in rigid steel conduit. The assumed diversified Power Factor is 0.95.

Step 1: Gather NEC Table 9 Constants
From NEC Table 9 (Copper, Steel conduit, 60Hz):
• R_ac for 6 AWG = 0.51 Ω/kft
• X_L for 6 AWG = 0.051 Ω/kft

Step 2: Calculate Reactive Factor (sinθ)
• cosθ = 0.95
• sinθ = √(1 - 0.95²) = √(1 - 0.9025) = √0.0975 = 0.312

Step 3: Calculate Effective Impedance (Z_eff)
• Z_eff = (0.51 × 0.95) + (0.051 × 0.312)
• Z_eff = 0.4845 + 0.0159 = 0.5004 Ω/kft

Step 4: Execute the Master Equation
• V_drop = 2 × 40A × (150ft / 1000) × 0.5004 Ω/kft
• V_drop = 80 × 0.15 × 0.5004
• V_drop = 12 × 0.5004 = 6.00 Volts

Step 5: Calculate Percentage
• %VD = (6.00V / 240V) × 100 = 2.50%
Result: Under the 3% threshold. 6 AWG Copper is correctly sized for this feeder.

Rearranged Forms: Solving for Length, Current, and Impedance

In field engineering, you rarely solve for V_drop directly. Usually, the voltage drop limit is fixed (e.g., 3.6V for a 120V circuit), and you need to find the maximum distance or the required wire impedance. Here are the algebraic rearrangements of the master equation.

💡 Tip: Define Z_eff First
For all rearranged forms below, let Z_eff = (R_ac × cosθ + X_L × sinθ). This simplifies the algebra significantly.

Solve for Maximum One-Way Length (L):
L = (V_drop × 1000) / (2 × I × Z_eff)
Use this to determine how far you can run a specific wire size before exceeding your voltage drop limit.
Solve for Maximum Load Current (I):
I = (V_drop × 1000) / (2 × L × Z_eff)
Use this when auditing an existing circuit to see how much additional load can be added safely.
Solve for Required Effective Impedance (Z_eff):
Z_eff = (V_drop × 1000) / (2 × I × L)
Use this to find the maximum allowable Ω/kft. Once calculated, scan NEC Table 9 to find the smallest wire gauge with a Z_eff lower than your result.

Wire Selection Decision Tree and Final Recommendation

Calculating the math is only half the job. The final step is translating that percentage into a physical purchasing decision. Use the following decision matrix to finalize your wire gauge based on your calculated %VD.

Calculated %VD	Circuit Type	Action Required
≤ 3.0%	Branch or Feeder	Accept. Wire size is optimal. Proceed with installation.
3.1% to 5.0%	Branch Circuit	Upsize. Increase wire gauge by one AWG step (e.g., 12 to 10).
3.1% to 5.0%	Feeder	Conditional Accept. Acceptable if the downstream branch circuit is short and total system VD remains under 5%.
> 5.0%	Any	Reject. Upsize by two AWG steps, parallel conductors, or increase system voltage.

The Concrete Default Pick

Let's terminate this analysis with a concrete resolution for the failing scenario in Example 1 (120V, 15A, 80ft, 12 AWG yielding 3.25% VD). Following the decision tree, a branch circuit exceeding 3.0% requires upsizing by one AWG step. Therefore, you must move from 12 AWG to 10 AWG.

Final Recommendation: Do not install 12 AWG for this run. The default pick is 10 AWG THHN Copper. Specifically, source Southwire Part #11582025 (or equivalent Cerrowire/Southwire 10 AWG THHN/THWN-2). Running the math on 10 AWG (R_ac = 1.24 Ω/kft in PVC) drops the V_drop to 2.48V (2.06%), safely clearing the 3% NEC threshold and ensuring your compressor motor starts without dimming the lights or tripping thermal overloads.