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. For information on testing three-phase kit, see three-phase testing.

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) offset in time. In this article angles will be measured in radians except where otherwise stated.

Variable setup and basic definitions

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Let

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

where $t$ is time and $f$ is frequency.

Using $x=ft$ the waveforms for the three phases are

$V\_\{L1\}=A\backslash sin\; x\backslash ,\backslash !$

$V\_\{L2\}=A\backslash sin\; \backslash left(x-\backslash frac\{2\}\{3\}\; \backslash pi\backslash right)$

$V\_\{L3\}=A\backslash sin\; \backslash left(x-\backslash frac\{4\}\{3\}\; \backslash pi\backslash right)$

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

Balanced loads

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Generally, in electric power systems the loads are 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

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

Using R = 1,

$P\_\{L1\}=\backslash frac\{V\_\{L1\}^\{2\}\}\{R\}=V\_\{L1\}^\{2\}\backslash ,\backslash !$

$P\_\{L2\}=\backslash frac\{V\_\{L2\}^\{2\}\}\{R\}=V\_\{L2\}^\{2\}\backslash ,\backslash !$

$P\_\{L3\}=\backslash frac\{V\_\{L3\}^\{2\}\}\{R\}=V\_\{L3\}^\{2\}\backslash ,\backslash !$

$P\_\{TOT\}=P\_\{L1\}+P\_\{L2\}+P\_\{L3\}\backslash ,\backslash !$

$P\_\{TOT\}=\backslash sin^\{2\}\; x+\backslash sin^\{2\}\; (x-\backslash frac\{2\}\{3\}\; \backslash pi)+\backslash sin^\{2\}\; (x-\backslash frac\{4\}\{3\}\; \backslash pi)$

Using angle subtraction formulae

$P\_\{TOT\}=\backslash sin^\{2\}\; x+\backslash left(\backslash sin\; x\backslash cos\backslash left(\backslash frac\{2\}\{3\}\; \backslash pi\backslash right)-\backslash cos\; x\backslash sin\backslash left(\backslash frac\{2\}\{3\}\; \backslash pi\backslash right)\backslash right)^\{2\}+\backslash left(\backslash sin\; x\backslash cos\backslash left(\backslash frac\{4\}\{3\}\; \backslash pi\backslash right)-\backslash cos\; x\backslash sin\backslash left(\backslash frac\{4\}\{3\}\; \backslash pi\backslash right)\backslash right)^\{2\}$

$P\_\{TOT\}=\backslash sin^\{2\}\; x+\backslash left(-\backslash frac\{1\}\{2\}\backslash sin\; x-\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash cos\; x\backslash right)^\{2\}+\backslash left(-\backslash frac\{1\}\{2\}\backslash sin\; x+\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash cos\; x\backslash right)^\{2\}$

$P\_\{TOT\}=\backslash sin^\{2\}\; x+\backslash frac\{1\}\{4\}\backslash sin^\{2\}\; x+\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash sin\; x\backslash cos\; x\; +\backslash frac\{3\}\{4\}\backslash cos^\{2\}\; x+\backslash frac\{1\}\{4\}\backslash sin^\{2\}\; x-\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash sin\; x\backslash cos\; x\; +\backslash frac\{3\}\{4\}\backslash cos^\{2\}\; x$

$P\_\{TOT\}=\backslash frac\{6\}\{4\}\backslash sin^\{2\}\; x+\backslash frac\{6\}\{4\}\backslash cos^\{2\}\; x$

$P\_\{TOT\}=\backslash frac\{3\}\{2\}(\backslash sin^\{2\}\; x+\backslash cos^\{2\}\; x)$

Using the Pythagorean trigonometric identity

$P\_\{TOT\}=\backslash frac\{3\}\{2\}$

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,

$I\_\{L1\}=V\_\{L1\}\backslash ,\backslash !$

$I\_\{L2\}=V\_\{L2\}\backslash ,\backslash !$

$I\_\{L3\}=V\_\{L3\}\backslash ,\backslash !$

$I\_\{N\}=I\_\{L1\}+I\_\{L2\}+I\_\{L3\}\backslash ,\backslash !$

$I\_\{N\}=\backslash sin\; x+\backslash sin\; \backslash left(x-\backslash frac\{2\}\{3\}\; \backslash pi\backslash right)+\backslash sin\; \backslash left(x-\backslash frac\{4\}\{3\}\; \backslash pi\backslash right)$

Using angle subtraction formulae

$I\_\{N\}=\backslash sin\; x+\backslash sin\; x\backslash cos\backslash left(\backslash frac\{2\}\{3\}\; \backslash pi\backslash right)-\backslash cos\; x\backslash sin\backslash left(\backslash frac\{2\}\{3\}\; \backslash pi\backslash right)+\backslash sin\; x\backslash cos\backslash left(\backslash frac\{4\}\{3\}\; \backslash pi\backslash right)-\backslash cos\; x\backslash sin\backslash left(\backslash frac\{4\}\{3\}\; \backslash pi\backslash right)$

$I\_\{N\}=\backslash sin\; x-\backslash frac\{1\}\{2\}\backslash sin\; x-\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash cos\; x-\backslash frac\{1\}\{2\}\backslash sin\; x+\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash cos\; x$

$I\_\{N\}=0\backslash ,\backslash !$

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

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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

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

Conversion to other phase systems

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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. The US Phase Converter Standards Organization conducts independent three phase tests on the various phase converter technologies and publishes the results.

If the frequency (HZ) of the three-phase power supplied does not match the frequency needed to run the machines or equipment a Frequency converter can be used.

References

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

Patents

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• -- "''Three-phase oscillator"

Electric power | Nikola Tesla

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