Three-phase circuits

Nearly all electric power is generated, transmitted and distributed as three-phase AC: three sinusoidal voltages of equal amplitude, displaced 120° from one another. A three-phase system delivers constant total power rather than the pulsating power of a single phase, carries the same power with less conductor material, and naturally produces the rotating magnetic field that drives induction and synchronous motors. Homes take a single-phase supply from the system, but commercial buildings, industrial plants and data centres are supplied and distributed in three phases.

This lesson describes balanced three-phase sources, wye and delta connections and the relations between line and phase quantities, the per-phase method for balanced systems, three-phase power and its measurement, and unbalanced loads and neutral currents.

Balanced three-phase voltages

A three-phase generator has three windings spaced 120° apart around its stator, so it produces three voltages that are equal in magnitude and 120° apart in phase. With phase a as reference and RMS phasors:

This is positive (abc) sequence: the voltages reach their peaks in the order a, b, c. Reversing any two phases gives negative (acb) sequence, which reverses the rotation of a three-phase motor; electricians check phase rotation before connecting motors for this reason. A key property of a balanced set is that the three phasors sum to zero: \(\mathbf{V}_{an} + \mathbf{V}_{bn} + \mathbf{V}_{cn} = 0\).

Wye and delta connections

Sources and loads can each be connected in two ways:

• Wye (Y, or star): one end of each phase is joined at a common neutral point; the other ends connect to the three lines. A fourth wire, the neutral, may be brought out.
• Delta (Δ): the three phases are connected end to end in a closed triangle, and the lines connect to the corners. There is no neutral.

Two kinds of quantity must be kept distinct. Phase quantities are those of an individual source winding or load impedance; line quantities are those on the lines, between lines (voltages) or in them (currents).

Wye: line voltage is √3 times phase voltage

In a wye, each line current equals its phase current. The line-to-line voltage is the difference of two phase voltages:

So in a balanced wye, \(V_L = \sqrt3\,V_p\), and the line voltages lead the corresponding phase voltages by 30°. This is why North American wye systems are named by two voltages:

• 208Y/120 V: \(120 \times \sqrt3 = 208\) V between lines, 120 V line to neutral, used in offices, schools and data centres for receptacles and small loads.
• 600Y/347 V: the standard Canadian system for commercial and industrial buildings, with motors connected at 600 V and lighting at 347 V line to neutral.

Delta: line current is √3 times phase current

In a delta, each phase voltage equals a line voltage. Each line current is the difference of two phase currents at a corner, and for a balanced load

so \(I_L = \sqrt3\,I_p\), with line currents lagging their phase currents by 30°.

Delta-wye load conversion

A balanced delta of impedances \(\mathbf{Z}_\Delta\) is equivalent, at its terminals, to a balanced wye of

which follows from the delta-wye transformation of ELE-202 with three equal impedances. Converting every delta to a wye allows any balanced system to be analysed by the per-phase method.

The per-phase method

In a balanced system, the three phases carry identical currents shifted by 120°, and the neutral currents cancel. The neutral point of a balanced wye load is therefore at the same potential as the source neutral whether or not a neutral wire is present, and one phase can be analysed alone:

• Convert any delta sources or loads to equivalent wyes.
• Draw phase a only, from the source neutral through the line impedance and load impedance back to neutral, using line-to-neutral voltage.
• Solve the single-phase circuit.
• Obtain phases b and c by shifting the results by −120° and +120°, and line or delta quantities from the √3 relations.

Three-phase power

Total power is three times the power per phase. In terms of the line quantities that can be measured directly, for either connection,

where \(\theta\) is the angle of the per-phase load impedance (the angle between phase voltage and phase current, not between line voltage and line current).

A remarkable property of a balanced three-phase system is that the total instantaneous power is constant: the \(2\omega\) pulsations of the three phases are 120° apart (240° at twice the frequency) and cancel. A three-phase motor therefore produces smooth torque, whereas a single-phase motor's torque pulsates at 120 Hz.

Three-phase transmission also saves conductor material. To deliver power \(P\) at line voltage \(V\) with the same total \(I^2R\) loss, a three-wire three-phase line needs 75% of the copper of a two-wire single-phase line at the same voltage between conductors.

Measuring three-phase power

Total power in a three-wire system can be measured with just two wattmeters (Blondel's theorem: an n-wire system needs n − 1 wattmeters). Each wattmeter's current coil is in one line and its voltage coil between that line and the third line. For a balanced load with impedance angle \(\theta\),

and \(\tan\theta = \sqrt3\,(W_1 - W_2)/(W_1 + W_2)\), so the two readings also give the power factor. If the power factor is below 0.5 (\(\theta > 60°\)), one reading becomes negative. Modern digital power meters use the same principle with electronic sampling, and four-wire systems use three elements.

Unbalanced loads and the neutral

Real systems are never perfectly balanced: single-phase loads, such as lighting and receptacles, are spread across the three phases as evenly as possible, but they switch on and off independently. In a four-wire wye system, each single-phase load sees its line-to-neutral voltage, and the neutral carries the phasor sum of the three line currents. That sum is much less than the arithmetic sum, and is zero when the loads are balanced.

Harmonics complicate this. Third-harmonic currents (180 Hz) in the three phases are 3 × 120° = 360° apart, that is, in phase, so instead of cancelling in the neutral they add. In buildings full of computers and LED lighting, the neutral current can exceed the phase currents, and neutrals are sized accordingly (the electrical codes require neutrals of such circuits to be counted as current-carrying conductors). Delta-wye transformers trap these triplen harmonics in their delta windings, as described in DCE-310.

Unbalanced systems in general are analysed by writing mesh or nodal equations for the full three-phase circuit, or in power systems by the method of symmetrical components (ELE-380), which decomposes any unbalanced set into balanced positive-, negative- and zero-sequence sets.

Lesson 8 continues with unbalanced systems: floating neutrals, unbalanced delta loads, voltage unbalance in motors and symmetrical components for fault analysis.