Three-Phase Circuits

Author: E.E. Kimberly

Fig. 9-1 (a) is a three-dimensioned representation of a magnetic field in which there are three revolving conductors displaced 120 degrees from one another on their common circular path.¹ The voltages generated in them will then be displaced in time by 120 degrees, as shown in Fig. 9-1 (b) by the sinusoids e₀₁, e₀₂, and e₀₃ and their corresponding vectors. The conductors will be said to have a positive sense from the back, or zero, end toward the nearer end, as indicated by the sequence of the numbers in the subscripts identifying the vectors which represent the voltages generated in them. It is common practice to connect the conductors or phases inside the machine, as at 0, so that the terminals 1, 2, and 3 only are necessary to connect the machine to its load. The generator is then said to be Y-connected or wye-connected.

A voltmeter connected to the terminals 1 and 2 in Fig. 9-1 (a) would be affected by the voltages E₀₁ and E₀₂. However, if the circuit be traced through the two conductors from terminal 1 to terminal 2, the course would be in a negative sense through conductor 1 and in a positive sense through conductor 2. Therefore, the voltage appearing between terminals 1 and 2 will be the vector difference of E₀₁ and E₀₂, and not their vector sum. It is not important whether this vector difference be taken as E₀₁-E₀₂ or E₀₂-E₀₁. If, however, the voltage between terminals 1 and 2 is to be called E₁₂, and the double subscript is to be consistent in denoting the vector direction, it is necessary that E₁₂ = E₀₂ - E₀₁. Likewise, E₂₃ = E₀₃ - E₀₂ and E₃₁ = E₀₁ - E₀₃. The voltages E₀₁, E₀₂, and E₀₃ are called the phase voltages; and E₁₂, E₂₃, and E₃₁ are called the line-to-line voltages or simply the line voltages. The voltage at which a generator is rated is always the line voltage, unless otherwise stated. By the trigonometry of Fig. 9-1 (b),

Since E₀₁ = E₀₂ (in magnitude),

or

1	In practical generators many conductors are used in every phase. They are arranged in many ways not described in this book. For further study of windings, see "Theory of Alternating Current Machinery," by A. S. Langsdorf, McGraw-Hill Book Co.