Voltage drop in AC circuits

ELEC-110 introduced voltage drop as the product of current and conductor resistance. That is accurate for DC and a good approximation for small branch circuits, but in AC feeders, especially large ones, the conductor's reactance and the load's power factor change the result, sometimes substantially. This lesson develops the AC voltage drop calculation: the resistance and reactance of conductors and what affects them, the effective impedance method, single-phase and three-phase formulas, the code limits and tables, sizing conductors for voltage drop, and special cases such as motor starting and long runs to outbuildings, EV chargers and pumps.

Conductor impedance in AC circuits

An AC conductor has both resistance and inductive reactance:

• AC resistance is slightly higher than DC resistance, because of skin effect (alternating current crowds toward the surface of the conductor, using less of its cross-section) and proximity effect (currents in nearby conductors distort the current distribution). The difference is negligible for small conductors and a few percent for large ones such as 500 kcmil. Resistance also rises with temperature, by about 0.4% per °C for copper; tables are usually given at 75 °C.
• Inductive reactance arises from the magnetic field around the conductors and depends on their spacing: conductors close together (in a cable or tight conduit) have less reactance than widely spaced ones. Steel conduit increases reactance compared with PVC or aluminum conduit, because the steel concentrates the magnetic field. Reactance changes little with conductor size, typically 0.04 to 0.06 Ω per 1,000 ft (0.13 to 0.2 Ω/km) for building wiring.

For small conductors, resistance dominates and reactance can be ignored. For large conductors, the resistance becomes small and reactance becomes comparable or even larger, so ignoring it underestimates voltage drop.

The effective impedance method

The voltage drop along a conductor is the phasor \(I(R + jX)\). What matters for the load is how much this reduces the magnitude of the voltage, which, for the small drops of practical circuits, is closely approximated by the component of the drop in phase with the source voltage:

where \(\theta\) is the load's power factor angle (\(\cos\theta\) is the power factor). At unity power factor, only resistance matters. At lagging power factors, reactance contributes; for large conductors with X/R near or above 1, the effective impedance at 0.85 power factor can be more than the resistance alone, while for small conductors it is less (because \(\cos\theta < 1\) reduces the resistive part).

The voltage drop formulas are then:

where \(L\) is the one-way length and \(Z_{\text{eff}}\) the effective impedance per unit length. The factor 2 accounts for the outgoing and return conductors in a single-phase circuit; \(\sqrt3\) converts the drop in one line to the change in line-to-line voltage for a balanced three-phase circuit. For a three-phase line-to-neutral drop, use \(I\,L\,Z_{\text{eff}}\).

Code requirements and tables

CEC Rule 8-102 limits voltage drop in an installation: it must not exceed 5% from the supply side of the consumer's service (or the transformer secondary of a separately derived system) to the point of utilization, and must not exceed 3% in a feeder or branch circuit. The drop is calculated on the basis of the demand load of a feeder or, for a branch circuit, the connected load if known, or 80% of the rating of the overcurrent device protecting the circuit. Appendix D of the CEC includes tables (such as Table D3) giving circuit lengths for a 1% voltage drop for each conductor size and current, with correction factors for system voltage and type, which make routine checks quick.

Voltage drop matters for more than compliance. A motor running at 90% voltage draws more current, runs hotter and produces about 81% of its rated torque (torque varies with voltage squared); LED drivers and electronic equipment may shut down; resistive heaters lose output (with the square of voltage); and every volt lost is energy wasted as heat in the conductors.

Sizing conductors for voltage drop

To find the conductor needed for a given maximum drop, rearrange the formula for the allowable effective impedance per unit length:

and choose the smallest conductor whose effective impedance does not exceed it, then check that it also meets the ampacity requirements. For small conductors where reactance is negligible, the equivalent shortcut uses resistance only, or the circular-mil formula familiar from ELEC-110, \(\text{cmil} = k\,K\,I\,L/V_d\), with \(K\) about 12.9 Ω·cmil/ft for copper at 75 °C.

When conductors are increased in size to limit voltage drop, the bonding conductor is generally increased in proportion to the increase in circular mil area of the circuit conductors, so that fault current remains high enough to operate the protective device (Lesson 11).

Voltage drop with many loads and in distribution systems

A feeder supplying several loads along its length carries decreasing current toward its far end. Its total drop is the sum of the drops in each section, each calculated with the current flowing in that section. A convenient equivalent is to treat the total load as concentrated at the load centre, the current-weighted average distance.

In a complete installation, the voltage at a load is the transformer secondary voltage (set by taps) minus the transformer's own regulation under load (Lesson 8), minus the feeder drop, minus the branch circuit drop. Planning distribution so that large loads are near their sources, using higher voltages for long feeders (a 600 V or 480 V feeder carries the same power with a third or less of the current of a 208 V feeder, and the drop as a fraction of voltage falls with the square of the voltage ratio), and locating transformers near load centres are the most effective ways to control voltage drop.

Motor starting

Induction motors draw five to seven times their full-load current when starting, at low power factor (0.2 to 0.4). Voltage drop during starting can be large enough to stall the motor, drop out contactors (which may release below about 70% to 80% voltage) and flicker lights. For starting, the effective impedance uses the starting power factor, at which reactance dominates.

Measuring voltage drop in the field

Calculations predict voltage drop; measurements confirm it. The most direct method is to measure the voltage at the source (panel) and at the load at the same time, under the same load, with two meters or a meter with remote probes, because the supply voltage itself varies minute to minute. Alternatively, measure the voltage at the load with the load off and then on: the difference is the drop in the whole circuit, including the supply transformer, so it overstates the circuit drop slightly. A large drop on one conductor, measured end to end under load, points to a loose or corroded connection rather than an undersized conductor, especially if one conductor of a circuit shows much more drop than the other. Recording voltage at a critical load over a week with a power quality analyzer (Lesson 5) shows whether low voltage coincides with particular loads starting, which separates voltage drop in the building from low supply voltage.