Three-phase

This article deals with the basic mathematics and principles of three phase. For information on where, how and why Three phase is used please see three-phase electric power. For information on testing three phase kit please see three-phase testing.

Three phase systems have 3 waveforms (usually carrying power) that are 2/3π radians (120 degrees,1/3 of a cycle) offset in time. In this article angles will be measured in radians except where otherwise stated.

Contents

1 Variable setup and basic definitions

2 Balanced loads

2.1 Star connected systems with neutral

2.1.1 Constant power transfer
2.1.2 no neutral current

2.2 Star connected systems without neutral

3 Revolving Magnetic Field

4 Conversion to other phase systems

5 References

Variable setup and basic definitions

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Three_phase_voltages.png

One voltage cycle of a three-phase system, labelled 0 to 360 degrees ( 2 π radians) along the time axis. The plotted line represents the variation of instantaneous voltage (or current) with respect to time. This cycle will repeat 50 or 60 times per second depending on the power system frequency. The colours of the lines represent the American color code for three phase. That is black=V_L1 red=V_L2 blue=V_L3

Let

$x=2\pi ft\,\!$

where t is time and f is freqency.

Using x the waveforms for the three phases are

$V_{L1}=A\sin x\,\!$

$V_{L2}=A\sin (x-\frac{2}{3} \pi)$

$V_{L3}=A\sin (x-\frac{4}{3} \pi)$

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

Balanced loads

Generally, in electric power systems the load is distributed as evenly as 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.

To keep the calculations simple we shall normalise A and R to 1 for the remainder of these calculations

Star connected systems with neutral

Constant power transfer

Using R=1

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

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

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

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

$P_{TOT}=\sin^{2} x+\sin^{2} (x-\frac{2}{3} \pi)+\sin^{2} (x-\frac{4}{3} \pi)$

Using angle subtraction formulae

$P_{TOT}=\sin^{2} x+(\sin x\cos(\frac{2}{3} \pi)-\cos x\sin(\frac{2}{3} \pi))^{2}+(\sin x\cos(\frac{4}{3} \pi)-\cos x\sin(\frac{4}{3} \pi))^{2}$

$P_{TOT}=\sin^{2} x+(-\frac{1}{2}\sin x-\frac{\sqrt{3}}{2}\cos x)^{2}+(-\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)^{2}$

$P_{TOT}=\sin^{2} x+\frac{1}{4}\sin^{2} x+\frac{\sqrt{3}}{2}\sin x\cos x +\frac{3}{4}cos^{2} x+\frac{1}{4}\sin^{2} x-\frac{\sqrt{3}}{2}\sin x\cos x +\frac{3}{4}cos^{2} x$

$P_{TOT}=\frac{6}{4}\sin^{2} x+\frac{6}{4}\cos^{2} x$

$P_{TOT}=\frac{3}{2}(\sin^{2} x+\cos^{2} x)$

Using the Pythagorean trigonometric identity

$P_{TOT}=\frac{3}{2}$

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

The neutral current is the sum of the phase currents

Since R=1

$I_{L1}=V_{L1}\,\!$

$I_{L2}=V_{L2}\,\!$

$I_{L3}=V_{L3}\,\!$

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

$I_{N}=\sin x+\sin (x-\frac{2}{3} \pi)+\sin (x-\frac{4}{3} \pi)$

Using angle subtraction formulae

$I_{N}=\sin x+\sin x\cos(\frac{2}{3} \pi)-\cos x\sin(\frac{2}{3} \pi)+\sin x\cos(\frac{4}{3} \pi)-\cos x\sin(\frac{4}{3} \pi)$

$I_{N}=\sin x-\frac{1}{2}\sin x-\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x$

$I_{N}=0\,\!$

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 only generally used 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.

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 measure 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 auxilary functions the motor will not accelerate from a stop when energized.

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 degree 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.

References

William D. Stevenson Jr., "Elements of Power Systems Analysis", 3rd ed. 1975, McGraw Hill, New York USA ISBN 0070612854