If an electric current flows through a wire, it causes a magnetic field H as shown in Figure 5.18 for a wire and a coil.

The magnetic field strength H at a distance r from an active wire with electric current I is:

Figure 5.18 Magnetic Fields H Produced by an Electric Current in a

Wire and Coil

Figure 5.18 Magnetic Fields H Produced by an Electric Current in a

Wire and Coil

Figure 5.19 Left: Cross-section through a Stator with Three Coils Staggered by 120° for the Generation of a Rotating Field (Concentrated Winding); Middle: Cross-section; Right: Three-dimensional Drawing of an Integrated Rotating Field Winding (Distributed Winding)

Figure 5.19 Left: Cross-section through a Stator with Three Coils Staggered by 120° for the Generation of a Rotating Field (Concentrated Winding); Middle: Cross-section; Right: Three-dimensional Drawing of an Integrated Rotating Field Winding (Distributed Winding)

Besides the magnetic field strength, the magnetic induction B can be estimated using the magnetic permeability of free space n0:

and the material-dependent permeability coefficient nr from the following:

The magnetic field of active wires is used for generating a rotating field. Therefore, a three-phase AC flows through three coils that are staggered by 120°. Figure 5.19 shows three coils U, V and W that are staggered by 120° with the connections U1, U2, V1, V2, W1 and W2.

If three ACs, that are staggered temporally by 120°, are fed into 120° spatially staggered three-phase windings, a rotating field emerges. Figure 5.20 explains this for two points in time. The currents at stages I (mt = 0) and II (at = n/2) can be estimated from Figure 5.21. The currents create a magnetic field as explained above. The north pole of the magnetic field rotates clockwise by 90° between stages I and II. If the magnetic field is constructed in the same manner for other points in time, it can be realized that the magnetic field changes its direction continuously. This changing magnetic field is also called a rotating field.

A magnetic needle inside a stator with three-phase windings would follow the rotating field and therefore rotate continuously. The frequency of the current defines the rotational speed. Hence, the magnetic needle would make one rotation during every period of the current. At the European mains

frequency of f = 50 Hz, the synchronous speed nS, i.e. the speed of the rotating magnetic field, is nS = 50 s-1 = 3000 min-1. For a frequency p of 60 Hz, ns becomes 3600 min-1.

The magnetic field of the stator with three-phase windings in Figure 5.20 has only two poles N and S, i.e. it has one pole pair (p = 1). Stators and windings can also be produced with more pole pairs. When doubling the pole pairs the rotational speed halves if the mains frequency remains constant. The mains frequency f and the pole pair number p define the synchronous speed:

The pole pitch can be calculated with the pole pair number p and the diameter d of the stator:

For a distance x along the circumference, the temporal and spatial staggered magnetic induction B with the amplitude B becomes:

The six connections U1, U2, V1, V2, W1 and W2 of the three windings are connected either to a star or to a delta connection as shown in Figure 5.22. This reduces the number of external connections to the three phases L1, L2 and L3. These three phases are often also called R, S and T. The neutral conductor N can be added as reference; however, a neutral or zero conductor is not necessary for three-phase machines.

The rms values of the phase-to-phase voltages and the phase to neutral conductor voltages are related as follows:

In the European grid, the rms value of the voltage between phase and neutral conductors is 230 V and between two phases V3^230 V = 400 V. All voltages are phase-shifted by 2n/3.

For the star connection, the voltages VY at the windings are equal to the corresponding phase-to-neutral conductor voltages. The voltages VD at the windings of a delta connection are equal to the phase-to-phase voltages V. Hence, the connection between the rms voltages is:

The currents IY in the windings of a star connection are the same as in the phases I:

The phase currents I of the delta connection are split between the currents in the windings. This reduces the rms currents IA in the windings by a factor of V3 The result is:

Since the voltages VY at the windings of the star connection are reduced by the factor V3, the currents IY in the windings are also reduced by the factor V3 relative to the delta connection:

Thus, the phase currents I of a star and a delta connection with the same windings and the same phase-to-phase voltages are not identical.

The total active power of a symmetrical three-phase system in a star connection is given by the sum of the active powers of all three windings:

The total active power of a delta connection is calculated similarly:

The measurement of voltage and current of only one phase is therefore sufficient to estimate the total power of a symmetrical connection. With VA = V3 • VY and IA = V3 • IY, the correlation between the active power of a star and a delta connection becomes:

In other words, the power input of a delta connection is three times higher than the associated star connection.

The reactive power Q and the apparent power S of delta and star connections are calculated similarly to the active power. Hence, they are:

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