Introduction to Electrical Engineering  Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.) 
8. Threephase Current 

The explanations given in Section 8.1. show that threephase 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 threephase 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 Yconnection) and the delta connection.
The interlinking of the three phases of a threephase 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 (U_{L1N}, U_{L2N}, U_{L3N}) which is designated by U_{Str}, and between two phase conductors (U_{L1L2}, U_{L1L3} U_{L2L3}) which is designated by U_{L}. 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 U_{Str} and U_{L}, 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 (U_{A}) with respect to the reference point and point B has a negative voltage (U_{B}) with respect to the reference point. Between the points A and B is the difference of the voltages U_{A} and U_{B} related to the reference point.
U_{AB} = U_{A}  U_{B}
When we assume that U_{A} = 2 V and U_{B} =  2 V, then we have for U_{AB}
U_{AB} = +2 V  ( 2 V)
U_{AB} = 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 linetoline voltage. By comparison of magnitudes in a vector graph true to scale, we obtain as magnitude relation between phase voltage and linetoline voltage
U_{L} = 1.73 · U_{Str}
Fig. 8.9. Graphical determination of the linetoline voltage
a) Vector diagram of the phase
voltages
b) Vector diagram of the phase and
linetoline voltages
The numerical factor 1.73 is the root extracted from 3 and, consequently, the equation can be written as
_{}
where:
U_{L} 
linetoline voltage (or phasetophase voltage) 
U_{str} 
phase voltage or star voltage or voltage to neutral 
Fig. 8.10. Triangle for calculating
the magnitude relation between phase voltage and linetoline voltage
(U_{Str} = phase voltage, U_{L} = linetoline voltage)
The ralation of magnitude between U_{L }and U_{Str} 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
U_{L}/2 = U_{str} · 30°U_{L} = 2 · U_{str} · cos 30°
_{}
_{}
Example 8.2.
The phase voltage of a threephase current network is 220 V. Which linetoline voltage is available in this network?
Given:
U_{Str} = 220
To be found:
U_{L}
Solution:
_{}U_{L} = 1.73 · 220 V
U_{L} = 380 V
A linetoline 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 linetoline voltage of 380 V is available from the same network for industrial enterprises.
In practice, sometimes the phase relation between phase voltage and linetoline voltage is utilised. Thus, Fig. 8.9.b shows that, for example U_{L1L2} exhibits a phase shift of 90° with respect to U_{L3N}. In general, it holds that the linetoline 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 linetoline voltage occurs. But in the junctions, a current division is obtained. For the upper junction we have
I_{L1} = I_{WZ}  I_{UX}
When the three linetoline 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 linetoline current a relation is given similar to the relation of magnitude between phase voltage and linetoline voltage. With the same phase load we have
where:
I_{L} 
linetoline current (or phasetophase current) 
I_{str} 
phase current 
_{}
Fig. 8.11. Vector representation of
the currents
Fig. 8.12. Delta connection
In threephase 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 linetoline (phasetophase) voltage is available, however, the difference between linetoline 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 linetoline (phasetophase) 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 phaseto 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?