Three-Phase Voltage Drop Formula √3·I·(Rcosφ+Xsinφ)·L — Calculator & IEC 60364 Limits

Voltage drop is the reduction in voltage between the supply point and the load terminals caused by the impedance of the cable. Excessive voltage drop causes equipment malfunction — motors run slower and hotter, lighting dims, and electronic equipment may shut down. Every wiring standard specifies maximum permissible voltage drop limits, and cable sizing must verify compliance.

The voltage drop check is often the governing criterion for cable selection, particularly for:

• Cable runs longer than approximately 50 m for power circuits
• Lighting circuits where visible flicker must be avoided
• Motor circuits where starting voltage drop affects torque
• Low-voltage distribution (e.g., 230 V single-phase) where the absolute voltage is already low

ECalPro implements voltage drop calculation for both single-phase and three-phase circuits, using impedance data from the applicable standard's tables, and accounts for the full complex impedance (resistance + reactance) — not just resistance, which is a common simplification that produces incorrect results for larger conductor sizes.

For a balanced three-phase circuit, the line-to-line voltage drop is:

ΔU = √3 × I × L × (R·cosφ + X·sinφ)     — (Eq. 1)

Where:
  ΔU   = voltage drop (V)
  I    = design current (A)
  L    = one-way cable route length (m)
  R    = AC resistance of conductor per unit length (Ω/m)
  X    = reactance of conductor per unit length (Ω/m)
  cosφ = power factor of the load
  sinφ = √(1 - cos²φ)

The percentage voltage drop relative to the nominal supply voltage:

ΔU% = (ΔU / U_n) × 100     — (Eq. 2)

Where U_n = nominal line-to-line voltage (e.g., 400 V)

This formula derives from the phasor relationship between the sending-end and receiving-end voltages. The term (R·cosφ + X·sinφ) is the projection of the cable impedance onto the load current phasor — it captures the in-phase and quadrature components of the voltage drop simultaneously.

Important note on sign: For highly capacitive loads (leading power factor), the term X·sinφ can become negative, meaning the reactance component actually reduces the voltage drop. This occurs with power factor correction capacitor banks and is correctly handled by ECalPro.

For a single-phase circuit (line and neutral), the voltage drop is:

ΔU = 2 × I × L × (R·cosφ + X·sinφ)     — (Eq. 3)

The factor of 2 accounts for the current flowing through both the phase conductor and the neutral conductor (the go and return path). The percentage is calculated against the phase-to-neutral voltage:

ΔU% = (ΔU / U₀) × 100     — (Eq. 4)

Where U₀ = nominal phase-to-neutral voltage (e.g., 230 V)

For three-wire single-phase systems (e.g., North American 120/240 V split-phase), the formula uses the line-to-line voltage and removes the factor of 2 when the load is connected between the two line conductors.

DC circuits: For direct current, reactance is zero, simplifying the formula to:

ΔU = 2 × I × L × R     — (Eq. 5)

This is the appropriate formula for solar PV string cables, battery circuits, and DC distribution systems.

A common engineering shortcut is to ignore the cable reactance (X) and calculate voltage drop using only resistance (R). This is acceptable for small conductors where resistance dominates the impedance. However, as conductor size increases, the resistance decreases (more copper area) while the reactance remains relatively constant (determined by conductor spacing, not size). Above approximately 35 mm², the reactance becomes a significant fraction of the total impedance.

The crossover point — where X exceeds R — depends on the installation method and conductor spacing:

The practical impact is significant. Consider a 240 mm² single-core copper XLPE cable at 0.85 power factor:

R = 0.0754 mΩ/m   (at 90°C operating temperature)
X = 0.0860 mΩ/m   (flat spaced formation)

Using R only:    voltage drop factor = R × cosφ = 0.0641 mΩ/m
Using R and X:   voltage drop factor = R × cosφ + X × sinφ
                                     = 0.0641 + 0.0453
                                     = 0.1094 mΩ/m

Error from ignoring X: 70.6% underestimate!

Ignoring reactance for a 240 mm² cable at 0.85 PF would underestimate the voltage drop by 70%. This is not a minor rounding difference — it could result in selecting a cable two sizes too small. ECalPro always uses the full R+X formula, sourcing impedance data from the standard's tables (IEC 60364-5-52 Table B.52.6, AS/NZS 3008.1.1 Tables 30–31, BS 7671 Appendix 4).

Different standards specify different permissible voltage drop limits and different approaches to allocating the drop across the distribution system.

IEC 60364-5-52 (International):

IEC 60364-5-52 Annex G (informative) recommends:

Note that IEC 60364 Annex G is informative, not normative. National committees may adopt different limits. The limits apply from the origin of the installation (main switchboard) to the load terminals.

AS/NZS 3000:2018 (Australia/New Zealand):

Clause 3.6.2 is normative (mandatory) and specifies:

BS 7671:2018+A2 (United Kingdom):

Appendix 12, Table 12.1:

BS 7671 treats the limits as applying to each section independently, not as a cumulative total from the origin. This is a subtle but important difference from AS/NZS 3000.

Key difference: AS/NZS 3000 applies a cumulative 5% from the point of supply, meaning the engineer must allocate the voltage drop budget between consumer mains, submains, and final subcircuits. IEC and BS standards typically apply limits per section. ECalPro allows the user to specify the upstream voltage drop already consumed, so the remaining budget is correctly calculated for the circuit being sized.

Real distribution boards rarely have a single uniform power factor. A typical commercial switchboard might supply:

• Lighting at PF = 0.95 (leading, with electronic ballasts)
• General power at PF = 1.00 (resistive loads)
• Motor loads at PF = 0.80–0.85 (lagging)
• HVAC compressors at PF = 0.75–0.80 (lagging)

For a feeder cable supplying a distribution board with mixed loads, ECalPro calculates the resultant power factor using the aggregate active and reactive power:

P_total = Σ P_i                           — (Eq. 6)
Q_total = Σ Q_i = Σ (P_i × tanφ_i)       — (Eq. 7)
S_total = √(P_total² + Q_total²)          — (Eq. 8)
cosφ_resultant = P_total / S_total         — (Eq. 9)

Where:
  P_i = active power of load i (W)
  Q_i = reactive power of load i (VAr)
  φ_i = power factor angle of load i

This resultant power factor is then used in the voltage drop formula (Eq. 1) for the feeder cable. Using the worst-case individual load power factor would overestimate the voltage drop; using unity power factor would underestimate it for predominantly motor loads.

For final subcircuits supplying a single load, the load's actual power factor is used directly. ECalPro's input form allows specifying either the power factor or the kW and kVAr separately, and calculates the resultant automatically.

The voltage drop budget must be carefully allocated across the distribution system. A common design approach is:

Point of supply to MSB:     1.0–1.5% (consumer mains)
MSB to sub-DB:              1.0–2.0% (submains)
Sub-DB to final load:       1.5–3.0% (final subcircuit)
────────────────────────────────────────
Total:                      ≤ 5.0%

This allocation means that a final subcircuit is not entitled to the full 5% — it must use only the remaining budget after the upstream sections consume their share. This is where many manual calculations go wrong: engineers calculate the final subcircuit voltage drop in isolation and compare it against 5%, ignoring the upstream drop.

ECalPro addresses this by:

1. Allowing upstream drop specification: The engineer enters the known voltage drop of upstream sections (consumer mains + submains). ECalPro then calculates the remaining budget available for the circuit being sized.
2. Cascaded calculation: When a project contains multiple interconnected circuits, ECalPro can calculate the cumulative voltage drop from the point of supply through all sections to the final load.
3. Flagging marginal results: If the total cumulative voltage drop is between 4% and 5%, ECalPro displays an amber warning, indicating that while the result technically complies, there is minimal margin for load growth or cable ageing.

For lighting circuits, AS/NZS 3000 Clause 3.6.3 recommends (but does not mandate) limiting the voltage drop to 3% to avoid perceptible dimming. Modern LED drivers can tolerate wider voltage ranges than incandescent lamps, but the 3% recommendation remains good practice and is the default in ECalPro for lighting circuit types.

IEC 60364-5-52 Annex G Note 2 additionally permits higher voltage drops during motor starting, provided the steady-state drop complies with the normal limits. ECalPro separately reports the starting voltage drop for motor circuits using the motor's locked-rotor current (typically 6–8 × FLC).

Cable resistance increases with temperature. The voltage drop tables in standards provide resistance values at a specific reference temperature (typically 20°C for resistance measurement, but the operating temperature for voltage drop). ECalPro corrects the resistance to the actual cable operating temperature using:

R_t = R_20 × [1 + α × (T_op - 20)]     — (Eq. 10)

Where:
  R_t  = resistance at operating temperature (Ω/m)
  R_20 = resistance at 20°C (Ω/m)
  α    = temperature coefficient of resistance
         (0.00393 /°C for copper, 0.00403 /°C for aluminium)
  T_op = cable operating temperature (°C)

The operating temperature depends on the load current relative to the cable's rated capacity. A cable loaded to 100% of its capacity operates at its maximum insulation temperature (75°C for PVC, 90°C for XLPE). A cable loaded to 50% operates at a significantly lower temperature. IEC 60364-5-52 Clause B.52.4 provides the formula for estimating the operating temperature based on load factor, and ECalPro uses this to provide a more accurate (and less conservative) voltage drop result.