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close this bookIntroduction to Electrical Engineering - Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.)
close this folder8. Three-phase Current
View the document8.1. Generation of Three-phase Current
View the document8.2. The Rotating Field
View the document8.3. Interlinking of the Three-phase Current
View the document8.4. Power of Three-phase Current

8.3. Interlinking of the Three-phase Current

The explanations given in Section 8.1. show that three-phase current involves three phases with two connections each. Consequently, six lines would be necessary as connection between generator and consumer or load. Such an open three-phase system ins not used in practice. By a combination of certain conductors, which is termed as interlinking, connecting lines can be saved. In practice two of such interlinking connections are used, namely, the star connection (also known as Y-connection) and the delta connection.

The interlinking of the three phases of a three-phase current into a star connection is shown in Fig. 8.7. For this purpose, the ends of the three generator coils (X, Y and Z) are connected and, in most cases, the connecting point is brought out as neutral conductor N. Then, four lines are required between generator and consumer.

Fig. 8.7. Star connection

When the load, resistors also arranged, in star connection have the same resistance values and cause an equal phase angle between current and voltage (in Fig. 8.7., pure active load, with a phase angle of 0 is represented.), then a line diagram will be brough about for the three currents which resembles that shown in Fig. 8.5. When we take a closer look at the sum of the three instantaneous values at any point of time in this line diagram, we will find that this sum is always equal to 0. Let us study this fact with particular respect to certain points of time on the basis of the line diagram shown in Fig. 8.5.a. For example, at point of time b, the phases L1 and L3 have half of the positive peak value and phase L2 the negative peak value. At point of time c, phase L3 has the value of 0, and the values of phases L1 and L2 are equal but of opposed directions. From this follows that no current is flowing in the neutral conductor R with the same phase load. In practice, equal phase load is seldom given so that a current flows in the neutral conductor which, however, is smaller than the current in the phase conductors L1, L2 and L3.

Fig. 8.7. also shows that a voltage can be tapped between phase conductor and neutral conductor (UL1N, UL2N, UL3N) which is designated by UStr, and between two phase conductors (UL1L2, UL1L3 UL2L3) which is designated by UL. The voltage between the phase conductors is produced by series connection of two generator coils. In order to be in a position to calculate the magnitude relations between UStr and UL, a similar circuit in a direct current circuit will be considered as a repetition (Fig. 8.8.).

Fig. 8.8. Determination of the voltage between the points A and B

Two voltage sources are connected in series and the point of connections is considered as the reference point (e.g. frame or earth). In this example, point A has a positive voltage (UA) with respect to the reference point and point B has a negative voltage (UB) with respect to the reference point. Between the points A and B is the difference of the voltages UA and UB related to the reference point.


When we assume that UA = 2 V and UB = - 2 V, then we have for UAB

UAB = +2 V - (- 2 V)
UAB = 4 V

Similar conditions are given in a star connection; however, the phase shift between the voltages sources connected in series must be taken into consideration. In order to ascertain the voltage between the phase conductors, we first have to represent the three phase voltages (also known as star voltage) with their phase shift of 120° to each other in the form of voltage vectors.

In order to be in a position to determine the voltage between the phase conductors, two voltages related to the neutral conductor as reference point in each case must be subra... l from each other by adding a further vector to the tip of a given vector but having opposite direction. As resultant vector we obtain the line-to-line voltage. By comparison of magnitudes in a vector graph true to scale, we obtain as magnitude relation between phase voltage and line-to-line voltage

UL = 1.73 · UStr

Fig. 8.9. Graphical determination of the line-to-line voltage

a) Vector diagram of the phase voltages

b) Vector diagram of the phase and line-to-line voltages

The numerical factor 1.73 is the root extracted from 3 and, consequently, the equation can be written as




line-to-line voltage (or phase-to-phase voltage)


phase voltage or star voltage or voltage to neutral

Fig. 8.10. Triangle for calculating the magnitude relation between phase voltage and line-to-line voltage (UStr = phase voltage, UL = line-to-line voltage)

The ralation of magnitude between UL and UStr can also be determined mathematically. For this purpose, Fig. 8.10. shows a triangle as a part of the representation given in Fig. 8.9. Using the trigonometric functions, we have

UL/2 = Ustr · 30°

UL = 2 · Ustr · cos 30°

Example 8.2.

The phase voltage of a three-phase current network is 220 V. Which line-to-line voltage is available in this network?


UStr = 220

To be found:



UL = 1.73 · 220 V

UL = 380 V

A line-to-line voltage of 380 V is available.

Example 8.2. demonstrates a typical case. With such a network, the voltage of 220 V desired by households can be supplied as the phase voltage while a line-to-line voltage of 380 V is available from the same network for industrial enterprises.

In practice, sometimes the phase relation between phase voltage and line-to-line voltage is utilised. Thus, Fig. 8.9.b shows that, for example UL1L2 exhibits a phase shift of 90° with respect to UL3N. In general, it holds that the line-to-line voltage has a phase shift of 90° with respect to the phase voltage of the phase conductor not under consideration.

In the delta connection shown in Fig. 8.11., one starts from the consideration that the sum of the individual voltages is always 0 at any time. Therefore, no balance current can flow within the generator in delta connection. In this circuit, only the line-to-line voltage occurs. But in the junctions, a current division is obtained. For the upper junction we have


When the three line-to-line currents have the same intensity, then the vector representation given in Fig. 8.12. is obtained. For the relation of magnitude between phase current and line-to-line current a relation is given similar to the relation of magnitude between phase voltage and line-to-line voltage. With the same phase load we have



line-to-line current (or phase-to-phase current)


phase current


Fig. 8.11. Vector representation of the currents

Fig. 8.12. Delta connection

In three-phase systems, two methods of interlinking are possible - the star connection and the delta connection. In the star connection, voltages can be tapped both between the phase conductors and between the neutral conductor and the phase conductors. In the delta connection, only the line-to-line (phase-to-phase) voltage is available, however, the difference between line-to-line current and phase current must be taken into consideration.

Questions and problems:

1. Sketch the star connection and the delta connection of the three coils of a generator!

2. Determine the relations of magnitude between phase voltage (voltage to neutral) and line-to-line (phase-to-phase) voltage by means of sketch true to scale! (Note: Select a length of 50 MM for the phase voltage!)

3. Calculate the voltage to neutral with a phase-to phase voltage of 220 V!

4. What is the value of the current flowing in the coils of a generator in delta connection when the current in the phase conductors is 34.6 A?