Why Is Three-Phase Power Used? The Real Reasons • The Science 360

Put your hand on a single-phase fan motor and you can feel a faint 100 Hz buzz (120 Hz in North America). That is not a fault. Power in a single-phase circuit really does drop to zero 100 times a second and climb back to twice its average in between, and the motor's torque follows it. A three-phase motor of the same size is noticeably smoother, and the reason is a small piece of trigonometry: three sine-squared waves offset by 120° add up to a perfect constant.

The pages that rank for this question usually lead with a different reason: "three-phase uses less copper." That is true, but the famous percentages (75%, "three times the power with one extra wire") each assume a particular comparison, and change the comparison and the number changes. One fair comparison makes three-phase no better than the split-phase system in a North American house. The things three phases do that nothing simpler can are the constant power and the rotating field. Here are all three, with the numbers.

Quick answerThree phases 120° apart deliver constant total power instead of power pulsing at twice mains frequency, they create a rotating magnetic field that runs induction motors without starting aids, and a balanced load returns no current in the neutral, so the same power needs less conductor than a single live-and-neutral circuit. Three is the fewest phases that does all three.

Infographic: Why is three-phase power used? A motor stator with three pairs of coils, 120 degrees apart, drives a magnetic field that rotates. Headline: three phases deliver constant power; each phase pulses 100 times a second but the three add to a flat line. The voltage between lines is 398.4 V, 230 V times root 3, and phase currents of 30, 20 and 10 A leave 17.3 A in the neutral. A chart over one cycle shows single-phase power swinging between zero and twice its average while the three-phase total stays constant. Tiles: a 3-phase rectifier never drops below 86.6% of peak, 221 kW was sent 176 km in 1891, and third harmonics can push neutral current to 173%. — Tap the image to view it full size. View the accessible, editable SVG.

Why single-phase power pulses and three-phase does not

Feed a resistor from $v = V_p\sin\omega t$ and the power is $v^2/R = (V_p^2/R)\sin^2\omega t$ . Sine squared never goes negative, but it touches zero twice per cycle. Now add two more phases shifted by 120° and 240° and sum the three powers:

\begin{aligned}\sin^2\theta &= \tfrac12(1 - \cos 2\theta) \\ \Sigma &= \tfrac12[3 - \cos 2\theta \\ &\quad - \cos(2\theta - 240^\circ) \\ &\quad - \cos(2\theta + 240^\circ)]\end{aligned}

The three cosines are 240° apart, which is the same as three phasors 120° apart: they sum to zero. So the sum is 3/2 at every instant, and:

p_{\mathrm{total}}(t) = \tfrac32 V_p^2/R = 3V_{\mathrm{rms}}^2/R

That is constant.

Three stacked plots over 40 milliseconds at 50 hertz. Top: three phase voltages 120 degrees apart. Middle: the power from each phase, a sine squared pulse between 0 and 1 at 100 hertz. Bottom: the three-phase total power is flat at 1.5 per unit, while a single-phase supply with the same average swings between 0 and 3. — Figure 1. Instantaneous power for a balanced resistive load. Each phase pulses at twice mains frequency; the three pulses interleave so precisely that the total is exactly constant.

Why does constant power matter? A generator turbine or a motor shaft must carry torque equal to power divided by speed. With single-phase, that torque pulsates at 100 or 120 Hz, which shakes bearings, mountings and windings, and is part of why single-phase motors hum (the same 100 Hz you hear from a humming transformer, though there the cause is magnetostriction). A balanced three-phase machine sees steady torque. The Wikipedia summary of three-phase power puts it as: "Power transfer into a linear balanced load is constant, which, in motor/generator applications, helps to reduce vibrations."

The same smoothing shows up when you rectify. A single-phase bridge gives two pulses per cycle that fall to zero; a three-phase bridge gives six that never fall below 86.6% of peak.

As fractions of the peak:

\begin{aligned}\text{3-phase min} &= \cos 30^\circ = \mathbf{0.866} \\ \text{3-phase avg} &= 3/\pi = \mathbf{0.955} \\ \text{1-phase avg} &= 2/\pi = \mathbf{0.637}\end{aligned}

The 3-phase bridge has 13.4% ripple before any capacitor.

Rectifier output with no smoothing capacitor over 40 milliseconds. The single-phase full-wave bridge output falls to zero every 10 milliseconds and averages 0.636 of peak. The three-phase six-pulse bridge output ripples between 0.866 and 1 of peak at 300 hertz and averages 0.955. — Figure 2. Bridge rectifier outputs normalised to peak. The capacitor a single-phase supply needs to fill those gaps is the subject of rectifier filter capacitor ripple; a three-phase bridge needs far less.

How three fixed coils make a rotating field

The second reason is what made three-phase win in the 1890s. Wind three coils around a motor's stator with their axes 120° apart in space, and drive them with currents 120° apart in time. Each coil's field only pulses back and forth along its own axis. Add the three as vectors and the result has constant length and turns smoothly at the supply frequency.

At θ = 0, coil L1 gives cos 0° = +1 along its axis; L2 and L3 give cos(−120°) = cos(120°) = −0.5 along theirs. Resolve along the L1 axis:

\begin{aligned}&1 + 2 \times (-0.5) \times \cos 120^\circ \\ &= 1 + 2(-0.5)(-0.5) = \mathbf{1.5}\end{aligned}

The perpendicular parts of L2 and L3 cancel. Repeat at any θ: magnitude 1.5, direction θ.

Left: three coil axes labelled L1, L2 and L3, 120 degrees apart. At time zero the L1 contribution is plus 1 along its axis and the L2 and L3 contributions are minus 0.5 each; drawn tip to tail they sum to a resultant of 1.5 along the L1 axis. Grey arrows show the resultant every 30 degrees of the cycle, all the same length on a circle. Right: field magnitude versus phase angle, constant at 1.5 for three phases and a cosine pulsing between plus and minus 1 for a single coil. — Figure 3. Three stationary coils, one rotating field of constant magnitude (computed). A single coil produces a field that pulses but has no direction of rotation, which is why a single-phase induction motor needs a start capacitor or shaded pole to get going.

Put a conducting cage in that rotating field and the induced currents drag it round: the induction motor, which has no brushes and no commutator and is still the workhorse of industry. Wikipedia notes that the rotating field of "specified direction and constant magnitude" means "no starting circuit is required". Reverse any two phases and the field, and the motor, runs backwards. A two-phase supply (Ferraris's original motor was two-phase) can also make a rotating field, but its common wire carries about 1.4 times the phase current, and three-phase displaced it.

Does three-phase really use 75% of the copper?

This is where most explanations get loose. The question is: for the same power, the same length and the same I²R loss, how much conductor do you need? The answer depends entirely on which voltage you hold equal, because insulation, safety and equipment are rated by voltage. Here is the derivation for the most quoted case, the same line-to-line voltage V:

Single phase, 2 wires, and three phase, 3 wires:

\begin{aligned}I_1 &= P/V,\quad \text{loss} = 2I_1^2 R_1 = \frac{2P^2 R_1}{V^2} \\ I_3 &= \frac{P}{\sqrt3\,V},\quad \text{loss} = 3I_3^2 R_3 = \frac{P^2 R_3}{V^2}\end{aligned}

Equal loss means R₃ = 2R₁, so each three-phase wire has half the cross-section. Copper: 3 × ½ = 1.5 units against 2 units, or 75%.

That matches the 0.75 figure on Wikipedia, which is careful to say "at the same phase-to-phase voltage". But a 230/400 V system is not compared with a 400 V single-phase circuit in practice; it is compared with an ordinary 230 V live and neutral. Hold the voltage to earth at 230 V instead and three-phase looks far better, at 25% (33% with a full-size neutral). Then compare it with a centre-earthed single-phase pair, where each wire sits at 230 V to earth on opposite sides of zero, and the advantage vanishes: that also needs 25%.

Bar chart of conductor volume for equal power, loss and length, relative to a single-phase live and neutral circuit at 100 percent. Three-phase three-wire: 75 percent at the same line-to-line voltage, 25 percent at the same voltage to earth. Three-phase four-wire with full neutral: 33.3 percent. Split-phase two-wire with an earthed centre: 25 percent. Split-phase three-wire with neutral: 37.5 percent. — Figure 4. Our calculation of relative conductor volume. Three-phase is not uniquely thrifty on a voltage-to-earth basis; it ties with a centre-earthed split-phase pair. Its edge over split-phase is the constant power and rotating field, plus easy 3-wire transmission.

Source note The 25%, 33% and 37.5% values and the tie with centre-earthed split phase are our own derivation using equal voltage to earth, equal loss and identical wires. I checked them twice by algebra and numerically, but I did not find a textbook table I could open to confirm them. Older power-systems texts tabulate conductor volumes on several such bases; the owner may want to compare against one before publishing.

Worked example: 23 kW over 50 m with 3% loss. Copper resistivity 1.72 × 10⁻⁸ Ω·m, all at 230 V to earth.

Single phase 230 V: I = 23,000/230 = 100 A, allowed loss 690 W, two conductors.

\begin{aligned}R &= 690/(2 \times 100^2) = 0.0345\ \Omega \\ A &= \rho L/R = \frac{1.72{\times}10^{-8} \times 50}{0.0345} \\ &= \mathbf{24.9\ mm^2} \\ V_{\mathrm{Cu}} &= 2 \times 50\ \mathrm{m} \times 24.9\ \mathrm{mm^2} \\ &= 2490\ \mathrm{cm^3} \approx \mathbf{22\ kg}\end{aligned}

Three phase 230/400 V: I = 23,000/(3 × 230) = 33.3 A.

\begin{aligned}R &= 690/(3 \times 33.3^2) = 0.207\ \Omega \\ A &= \mathbf{4.15\ mm^2} \\ V_{\mathrm{Cu}} &= 4 \times 50 \times 4.15 = 830\ \mathrm{cm^3} \\ &\approx \mathbf{7.4\ kg}\end{aligned}

That is four conductors (full neutral), a third of the single-phase copper.

Real cables come in standard sizes and are also limited by heating, but the ratio holds. Voltage drop scales the same way, as the wire gauge for 12 V post shows at the other end of the voltage range, and it is the same logic that pushes transmission to high voltage.

Why is three-phase 400 V and single-phase 230 V?

Both come from the same supply. Each line sits 230 V from neutral, and the lines are 120° apart, so the voltage between two lines is the third side of a triangle with two 230 V sides and a 120° angle.

\begin{aligned}V_{LL} &= 2 \times 230 \times \sin 60^\circ = 230\sqrt3 \\ &= \mathbf{398.4\ V}\ (\text{nominal } 400) \\ 120\sqrt3 &= 207.8 \to 208\ \mathrm{V} \\ 277\sqrt3 &= 479.8 \to 480\ \mathrm{V}\end{aligned}

Figure 5. Where the √3 comes from. Under IEC 60038, 230/400 V replaced the older 220/380 V and 240/415 V systems, with countries moving within ±10% of the new value by 2003.

The same √3 shows up in the power formula, which confuses people. Three-phase power is $P = \sqrt3\,V_{LL} I\cos\varphi$ , which is just $3V_{\mathrm{phase}} I\cos\varphi$ written with line voltage. It does not mean three-phase carries 1.73 times "more" than something.

A 32 A, 400 V three-phase supply at power factor 0.9:

\begin{aligned}P &= \sqrt3 \times 400 \times 32 \times 0.9 = \mathbf{19.95\ kW} \\ &\approx 3 \times 230 \times 32 \times 0.9 = 19.87\ \mathrm{kW}\end{aligned}

The second line uses the exact 230 V. A 32 A, 230 V single-phase circuit at unity power factor gives $230 \times 32 = \mathbf{7.36\ kW}$ .

What does the neutral carry?

In a perfectly balanced load, nothing: three equal currents 120° apart sum to zero, which is why high-voltage transmission lines need only three conductors. Homes are not balanced, so a 4-wire distribution feeder carries the phasor sum of the imbalance.

Phase currents 30 A, 20 A, 10 A (unity power factor):

\begin{aligned}I_N &= \sqrt{a^2 + b^2 + c^2 - ab - bc - ca} \\ &= \sqrt{1400 - 1100} \\ &= \sqrt{300} = \mathbf{17.3\ A}\end{aligned}

There is a modern twist. Rectifier loads (computers, LED drivers, phone chargers) draw current in short pulses rich in the third harmonic. Shifting a fundamental by 120° shifts its third harmonic by 360°, so the three phases' third harmonics are exactly in phase and add in the neutral instead of cancelling. Ian Evans, writing in the IET's Wired In, notes this "can result in up to 173% of phase current levels, even if all phases are completely balanced", which is why office buildings sometimes have oversized neutrals.

Why three phases, not two or six?

Constant power is not unique to three: any balanced set of N ≥ 3 symmetric phases has it, and so does two-phase at 90° ( $\sin^2 + \cos^2 = 1$ ). The trade-offs pick three:

Two phases need four wires to be independent, or three with a shared wire carrying √2 times the phase current, so they lose the neutral-cancellation trick. Six or twelve phases deliver smoother rectified DC, but every extra phase is another conductor, insulator string, breaker pole and transformer winding, for a ripple improvement that only very large rectifiers need. Those make their six or twelve pulses locally from three-phase with phase-shifting transformers. Three is the smallest number that gives constant power, a rotating field and zero neutral current with only three wires on the pylon.

History agreed quickly. Wikipedia credits Mikhail Dolivo-Dobrovolsky of AEG with a three-phase generator and motor in 1888, and the 1891 Lauffen to Frankfurt demonstration sent 221 kW at 15 kV over 176 km. The frequency those systems settled on is its own story, told in why AC is 50 or 60 Hz. And the reason a bird on one of those three wires is safe, while a hand bridging two would not be, is in why birds don't get electrocuted.

If you drive motors from DC instead, the electronics have to recreate those phases with switches; the single-phase version of that problem is the H-bridge motor driver, and a three-phase inverter is three half-bridges. At the high-current end, conductor size is also limited by the skin effect.

The short version: three sine-squared waves 120° apart sum to exactly 3/2, so three-phase power is constant and motors get steady torque. Three coils 120° apart make a field of constant strength that rotates, so induction motors start by themselves. The copper saving is real against a 230 V live and neutral (about a third of the copper with a full neutral) and 75% against single phase at the same line voltage, but a centre-earthed split-phase pair matches it. And 400 V is just 230 V × √3.

FAQ

Why is three-phase power used?

Three balanced phases deliver constant total power instead of power that pulses at twice the mains frequency, they create a rotating magnetic field that starts and drives induction motors without extra parts, and they move a given power with less conductor than an ordinary live-and-neutral circuit. Three is the smallest number of phases that gives all of this with no current in the neutral.

Why is three-phase 400 V and not 690 V?

The 400 V is the voltage between two lines, while 230 V is the voltage from each line to neutral. Two 230 V phasors 120° apart differ by $230\sqrt3 = 398\ \mathrm{V}$ , which is rounded to the nominal 400 V of IEC 60038. In North America the same factor gives 120/208 V and 277/480 V.

Does three-phase use less copper than single-phase?

It depends on what you hold equal. At the same line-to-line voltage, a three-wire three-phase circuit needs 75% of the conductor of a two-wire single-phase circuit for the same power and loss. At the same voltage to earth it needs 25%, or 33% with a full-size neutral. A centre-earthed split-phase circuit does as well on that basis.

Why not use six or twelve phases?

Three phases already give constant power, a rotating field and zero neutral current in a balanced load. More phases need more conductors, switchgear and transformer windings for little extra benefit. Six and twelve pulse arrangements are used inside large rectifiers, made from three-phase supplies with transformers.

Can the neutral current in a three-phase system be larger than the phase current?

Yes. Balanced sinusoidal currents cancel in the neutral, but third-harmonic currents from rectifier loads such as computers and LED drivers are in phase on all three lines and add. Harmonics specialists report neutral currents up to about 173% of the phase current in balanced systems with many such loads.

Sources

Wikipedia, "Three-phase electric power" (0.75 conductor at same phase-to-phase voltage, constant power into balanced load, rotating field with no starting circuit, neutral and triplen harmonics, two-phase common wire about 1.4 times, Dolivo-Dobrovolsky 1888, Lauffen to Frankfurt 1891 221 kW at 15 kV over 176 km): en.wikipedia.org/wiki/Three-phase_electric_power
Wikipedia, "IEC 60038" (230/400 V replacing 220/380 V and 240/415 V, Amendment 1 of 1994, ±10% by 2003, 120/208, 277/480 and 120/240 V systems): en.wikipedia.org/wiki/IEC_60038
Ian Evans, "Triplen harmonics" solution article, Wired In (IET), March/April 2007, hosted by Mirus International (neutral current up to 173% of phase current): mirusinternational.com PDF
Electricity Forum, "3 Phase Power Explained" (typical page-one explanation): electricityforum.com/3-phase-power

Method note: all five figures are computed from the equations in the text with numpy. The conductor comparisons assume balanced resistive loads, identical conductors within each system and equal I²R loss; they are our calculations.

Header image: NASA (image GRC-1979-C-02443), regraded for this article.