Definitions

# Three-phase

[three-feyz]
This article deals with the basic mathematics and principles of three-phase electricity. For information on where, how and why three-phase is used, see three-phase electric power.

In electrical engineering, three-phase electric power systems have at least three conductors carrying voltage waveforms that are 2π/3 radians (120°,1/3 of a cycle in-phase) offset in time. In this article angles will be measured in radians except where otherwise stated.

## Symbol

Also written as "phi"

## Variable setup and basic definitions

Let $x$ be the instantaneous phase of a signal of frequency $f$ at time $t$:

$x=2pi ft,!$

Using this, the waveforms for the three phases are

$V_\left\{L1\right\}=V_Psin x,!$

$V_\left\{L2\right\}=V_Psin left\left(x-frac\left\{2\right\}\left\{3\right\} piright\right)$

$V_\left\{L3\right\}=V_Psin left\left(x-frac\left\{4\right\}\left\{3\right\} piright\right)$

where $V_P$ is the peak voltage and the voltages on L1, L2 and L3 are measured relative to the neutral.

Generally, in electric power systems, the loads are distributed as evenly as is practical between the phases. It is usual practice to discuss a balanced system first and then describe the effects of unbalanced systems as deviations from the elementary case.

### Star connected systems with neutral

This refers to a system with a resistive load $R$ between each phase and neutral.

#### Constant power transfer

An important property of three-phase power is that the power available to a resistive load, $P = V I = frac\left\{V^2\right\}R$, is constant at all times.

$P_\left\{L1\right\}=frac\left\{V_\left\{L1\right\}^\left\{2\right\}\right\}\left\{R\right\},!$

$P_\left\{L2\right\}=frac\left\{V_\left\{L2\right\}^\left\{2\right\}\right\}\left\{R\right\},!$

$P_\left\{L3\right\}=frac\left\{V_\left\{L3\right\}^\left\{2\right\}\right\}\left\{R\right\},!$

$P_\left\{TOT\right\}=P_\left\{L1\right\}+P_\left\{L2\right\}+P_\left\{L3\right\},!$

To simplify the math, we define a nondimensionalized power for intermediate calculations, $p = frac\left\{P_\left\{TOT\right\} R\right\}\left\{V_P^2\right\}$

$p=sin^\left\{2\right\} x+sin^\left\{2\right\} left\left(x-frac\left\{2\right\}\left\{3\right\} piright\right)+sin^\left\{2\right\} left\left(x-frac\left\{4\right\}\left\{3\right\} piright\right)$

$p=sin^\left\{2\right\} x+left\left(sin xcosleft\left(frac\left\{2\right\}\left\{3\right\} piright\right)-cos xsinleft\left(frac\left\{2\right\}\left\{3\right\} piright\right)right\right)^\left\{2\right\}+left\left(sin xcosleft\left(frac\left\{4\right\}\left\{3\right\} piright\right)-cos xsinleft\left(frac\left\{4\right\}\left\{3\right\} piright\right)right\right)^\left\{2\right\}$

$p=sin^\left\{2\right\} x+left\left(-frac\left\{1\right\}\left\{2\right\}sin x-frac\left\{sqrt\left\{3\right\}\right\}\left\{2\right\}cos xright\right)^\left\{2\right\}+left\left(-frac\left\{1\right\}\left\{2\right\}sin x+frac\left\{sqrt\left\{3\right\}\right\}\left\{2\right\}cos xright\right)^\left\{2\right\}$

$p=sin^\left\{2\right\} x+frac\left\{1\right\}\left\{4\right\}sin^\left\{2\right\} x+frac\left\{sqrt\left\{3\right\}\right\}\left\{2\right\}sin xcos x +frac\left\{3\right\}\left\{4\right\}cos^\left\{2\right\} x+frac\left\{1\right\}\left\{4\right\}sin^\left\{2\right\} x-frac\left\{sqrt\left\{3\right\}\right\}\left\{2\right\}sin xcos x +frac\left\{3\right\}\left\{4\right\}cos^\left\{2\right\} x$

$p=frac\left\{6\right\}\left\{4\right\}sin^\left\{2\right\} x+frac\left\{6\right\}\left\{4\right\}cos^\left\{2\right\} x$

$p=frac\left\{3\right\}\left\{2\right\}\left(sin^\left\{2\right\} x+cos^\left\{2\right\} x\right)$

Using the Pythagorean trigonometric identity

$p=frac\left\{3\right\}\left\{2\right\}$

Hence (substituting back): $P_\left\{TOT\right\}=frac\left\{3 V_P^2\right\}\left\{2R\right\}$

since we have eliminated $x$ we can see that the total power does not vary with time. This is essential for keeping large generators and motors running smoothly.

#### No neutral current

For the case of equal loads on each of three phases, no net current flows in the neutral. The neutral current is the sum of the phase current.

$I_\left\{L1\right\}=frac\left\{V_\left\{L1\right\}\right\}\left\{R\right\},!$

$I_\left\{L2\right\}=frac\left\{V_\left\{L2\right\}\right\}\left\{R\right\},!$

$I_\left\{L3\right\}=frac\left\{V_\left\{L3\right\}\right\}\left\{R\right\},!$

$I_\left\{N\right\}=I_\left\{L1\right\}+I_\left\{L2\right\}+I_\left\{L3\right\},!$

We define a nondimensionalized current, $i=frac\left\{I_\left\{N\right\}R\right\}\left\{V_P\right\}$.

$i=sin x+sin left\left(x-frac\left\{2\right\}\left\{3\right\} piright\right)+sin left\left(x-frac\left\{4\right\}\left\{3\right\} piright\right)$

$i=sin x+sin xcosleft\left(frac\left\{2\right\}\left\{3\right\} piright\right)-cos xsinleft\left(frac\left\{2\right\}\left\{3\right\} piright\right)+sin xcosleft\left(frac\left\{4\right\}\left\{3\right\} piright\right)-cos xsinleft\left(frac\left\{4\right\}\left\{3\right\} piright\right)$

$i=sin x-frac\left\{1\right\}\left\{2\right\}sin x-frac\left\{sqrt\left\{3\right\}\right\}\left\{2\right\}cos x-frac\left\{1\right\}\left\{2\right\}sin x+frac\left\{sqrt\left\{3\right\}\right\}\left\{2\right\}cos x$

$i=0,!$

Hence also $I_N=0,!$

### Star connected systems without neutral

Since we have shown that the neutral current is zero we can see that removing the neutral core will have no effect on the circuit, provided the system is balanced. In reality such connections are generally used only when the load on the three phases is part of the same piece of equipment (for example a three-phase motor), as otherwise switching loads and slight imbalances would cause large voltage fluctuations.

## Unbalanced systems

Practical systems rarely have perfectly balanced loads, currents, voltages or impedances in all three phases. The analysis of unbalanced cases is greatly simplified by the use of the techniques of symmetrical components. An unbalanced system is analyzed as the superposition of three balanced systems, each with the positive, negative or zero sequence of balanced voltages.

## Revolving magnetic field

Any polyphase system, by virtue of the time displacement of the currents in the phases, makes it possible to easily generate a magnetic field that revolves at the line frequency. Such a revolving magnetic field makes polyphase induction motors possible. Indeed, where induction motors must run on single-phase power (such as is usually distributed in homes), the motor must contain some mechanism to produce a revolving field, otherwise the motor cannot generate any stand-still torque and will not start. The field produced by a single-phase winding can provide energy to a motor already rotating, but without auxiliary mechanisms the motor will not accelerate from a stop when energized.

## Visualization of the revolving magnetic field

As displayed in the image, when summarized, the three phase vectors of an ideal revolving magnetic field will form a circular pattern. In practical applications however, this may not always be the case. There can be technical malfunctions such as a missing phase, uneven phase voltage, or unbalanced loads without a neutral wire. This is unusual in the main grid, but sometimes happen on ships and other independent systems. It is also possible that the main grid is operating correctly, but that the phase voltages have been manipulated by an electronic motor control device. In these situations a vector visualization unit is used to display the actual form of the voltage-, current- and magnetic field vectors on a screen. For engineers this may be important information since any distortion from the ideal circular pattern will lead to vibrations, loss of power and risk for overload.

## Conversion to other phase systems

Provided two voltage waveforms have at least some relative displacement on the time axis, other than a multiple of a half-cycle, any other polyphase set of voltages can be obtained by an array of passive transformers. Such arrays will evenly balance the polyphase load between the phases of the source system. For example, balanced two-phase power can be obtained from a three-phase network by using two specially constructed transformers, with taps at 50% and 86.6% of the primary voltage. This Scott T connection produces a true two-phase system with 90° time difference between the phases. Another example is the generation of higher-phase-order systems for large rectifier systems, to produce a smoother DC output and to reduce the harmonic currents in the supply.

When three-phase is needed but only single-phase is readily available from the utility company a phase converter can be used to generate three-phase power from the single phase supply.

## References

• Stevenson, William D., Jr. (1975) Elements of Power Systems Analysis, McGraw-Hill electrical and electronic engineering series, 3rd ed., New York: McGraw Hill, ISBN 0-07-061285-4