Three-phase circuits of voltage supplies are the three connected voltage supplies, with the same magnitude of amplitude and frequency, but shifted to 120 grad. A typical three-phase circuit is depicted in Figure 1. The N shows the neutral point tied to ground.

Figure 1. Schematic depiction of a three-terminal.
Figure 1. Schematic depiction of a three-terminal.

There are various ways of load connection to the neutral point, depicted in Figure 2 (a,b,c). There are five simple types of connection of a three-phase generator to the three-phase load, depicted in Figure 3 (a-e).

Let’s think that positive current direction is from the source to the load. The wires, connecting the source and load are called linear. Currents and voltages through linear wires are called linear currents and linear voltages – IL and VL. Every generator coil is called a phase. Every load of the generator coil is called phase load. Currents through this load are called phase currents, voltages – phase voltages, Iph and Vph.

Figure 2. The examples of interconnections of three-phase circuits – the star and delta with or without neutral point.
Figure 2. The examples of interconnections of three-phase circuits – the star and delta with or without neutral point.
Figure 3. The standard connection schemes for three-phase source and load.
Figure 3. The standard connection schemes for three-phase source and load.

The phase and linear currents and voltage relationships depend on the generator connection scheme – star or delta. When the generator is a star connection, voltages are UL=3Uph.  When load is star connected currents are IL = Iph. When a generator is delta connected, and when a load is delta connected, the currents can be calculated by Kirchhoff Law. Figure 4 depicts the idea of the phase and linear voltages with the delta or star interconnection.

Figure 4. The schematic explanation of the phase and linear voltages relationship.
Figure 4. The schematic explanation of the phase and linear voltages relationship.

Let’s consider a few examples of the three-phase circuits calculation. We must remember that three-phase circuits are the circuits with sinus current. It is necessarily to keep in mind the operator of a symmetrical system a=1120,a2=1240. These two operators are moving a vector to 2π3 or 4π3 angle.

Figure 3a, UNN=EaYa+EbYb+EcYcYa+Yb+Yc=Ea(Ya+Yba+Yca2)Ya+Yb+Yc. If the load is distributed and Ya=Yb=Yc, then UNN=0, then UNN=0. If the load is not distributed the UNN0. and the voltages and currents of the delta branches can be calculated by Kirchhoff’s Law.

Active power of a three-phase circuit P is the sum of the active power of the phase load and active power of the neutral wire. The reactive power of the three-phase circuit S is the sum of reactive power of the phase load and a neutral wire. The total power can be seen as S=P2+Q2. To measure active power of the circuit with a neutral wire it should be used with three wattmeters; to measure active power without a neutral wire it should be used with two wattmeters.

When we put a sinusoidal current through a coil, the magnetic rotating field induces and concentrates around the coil core. The magnetic field vector has different directions in every single moment of time. When we have three coils, connected like a star with 2π3 angle shift between them, the sinusoidal current through the coils induces the circled rotating magnetic field.

If the magnitude of the current or the frequency through one of the coils is different, the inducing magnetic field is an elliptical rotating field. In circuits where coils are inducing the self-induction, it should be taken into account with the calculation we considered before.

#9 Periodic non-sinusoidal currents in linear circuits

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