11.2 Alternating Current, Single-Phase & Three-Phase Systems

11.2 Alternating Current, Single-Phase & Three-Phase Systems

Exam Focus: Frequency and RMS values, wye versus delta relationships, the 1.732 factor, converting between VA and amperes on single- and three-phase systems, and high-leg delta identification.

Every commercial load calculation on this examination ends with a division by either 240 (single-phase) or by 1.732 × line voltage (three-phase). Getting the system type right is worth more points than any single table lookup.

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Alternating Current Basics

• Frequency: North American power alternates at 60 hertz — 60 complete cycles per second, so the current reverses direction 120 times per second.
• Period: one cycle takes 1/60 second ≈ 16.67 milliseconds.
• RMS (effective) value: the DC-equivalent heating value. Every voltage and current in the NEC is an RMS value unless stated otherwise.

$E_{RMS} = 0.707 \times E_{PEAK} \qquad E_{PEAK} = 1.414 \times E_{RMS}$

A "120-volt" circuit has a peak of about 170 volts, and a "277-volt" circuit peaks near 392 volts. That matters for insulation ratings and for understanding why a 600-volt-rated conductor is used on a 480-volt system.

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Single-Phase Systems

On a 120/240 V system, the two ungrounded conductors are 180 degrees out of phase, which is why the voltage between them is the arithmetic sum, 240 V, and why the neutral of a properly connected multiwire branch circuit carries the difference of the two line currents.

$\text{Single-phase: } VA = E \times I \qquad I = \frac{VA}{E}$

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Three-Phase Systems and the 1.732 Factor

Three-phase power uses three ungrounded conductors whose voltages are 120 electrical degrees apart. The constant that connects line values to phase values is $\sqrt{3} = 1.732$.

+-------------------------------------------------------------------------+
|                  WYE vs. DELTA — MEMORIZE THIS BOX                      |
|                                                                         |
|   WYE (Y)                              DELTA                            |
|   ------------------------------       ------------------------------   |
|   E_line = 1.732 x E_phase             E_line = E_phase                 |
|   I_line = I_phase                     I_line = 1.732 x I_phase         |
|                                                                         |
|   Has a neutral point; supplies        No neutral unless one corner     |
|   line-to-neutral loads.               or midpoint is grounded.         |
|                                                                         |
|   208Y/120 V, 480Y/277 V               240 V, 480 V, 240/120 high leg   |
+-------------------------------------------------------------------------+

Why 208 Volts?

On a 208Y/120 V system each phase-to-neutral voltage is 120 V. The line-to-line voltage is:

$E_{line} = 1.732 \times 120 = 207.8 \approx 208\ \text{V}$

Likewise $1.732 \times 277 = 479.8 \approx 480\ \text{V}$.

[!CAUTION] The two-phases-of-a-wye trap. Two ungrounded conductors of a 208Y/120 V system are 120 degrees apart, not 180. A 240 V single-phase appliance will not work correctly on 208 V, and a resistance heater on 208 V produces only about 75% of its 240 V output, because power varies with the square of voltage: $(208/240)^2 = 0.751$.

Three-Phase Power Formulas

$VA = 1.732 \times E_{line} \times I_{line} \qquad I_{line} = \frac{VA}{1.732 \times E_{line}}$

Worked Example

A 208Y/120 V three-phase feeder carries a calculated load of 45,000 VA.

$I = \frac{45{,}000}{1.732 \times 208} = \frac{45{,}000}{360.3} = 124.9\ \text{A}$

The same 45,000 VA on a 480Y/277 V system draws only:

$I = \frac{45{,}000}{1.732 \times 480} = \frac{45{,}000}{831.4} = 54.1\ \text{A}$

That is the engineering reason commercial buildings distribute at 480 V and transform down: at 2.3 times the voltage, the conductor carries 2.3 times less current.

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The High-Leg Delta (120/240 V, 3-Phase, 4-Wire)

A four-wire delta system grounds the midpoint of one transformer winding so that two of the three phases give 120 V to ground for lighting, while all three give 240 V line-to-line for motors.

The third phase — the high leg, also called the wild leg or stinger — measures:

$E_{high\ leg} = 1.732 \times 120 = 208\ \text{V to the grounded conductor}$

[!IMPORTANT] Never land a 120-volt load on the high leg. A 120 V circuit connected between the high leg and the neutral sees 208 V and will destroy the load. Recognizing the 208 V reading on a 240 V delta system is a classic troubleshooting item.

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Quick Reference