## Equivalent circuits and circle diagrams for the stator current

The electrical structure of an asynchronous machine is similar to a transformer that consists of two coupled windings. The windings of an ideal transformer are coupled free of leakages. A real transformer has leakage fields and ohmic losses that can be considered by additional resistances and reactances in the equivalent circuit (see Figure 5.27).

Hence, the equations for the voltages of the transformer become:

With the number of turns per phase w1 and w2 and I

V2tr-1; R2 2 Xhas well as Xh1 become:

VV 1

X,, the voltage equations related to the primary side

In contrast to the transformer, the rotor winding of an asynchronous machine, which corresponds to the secondary windings of the transformer, is short-circuited (V'2 = 0). The rotor frequency f2 of an asynchronous machine is different from the frequency f1 of the stator. The slip s of the machine connects both frequencies:

The reactances of the rotor circuit become:

If X'2a and Xh in the voltage equations of the secondary site are replaced by s ' X'2a and s ■ Xh, respectively, and V'2 = 0 is used, the equation for the rotor becomes:

The splitting

provides the equivalent circuit for one phase of the asynchronous machine (see Figure 5.28).

With X'2 = X'2o + Xh and the voltage equation of the rotor, the current becomes:

Substituting the current and X1 = X1 stator and solving for I1 provides:

a + Xh into the voltage equation of the

This equation describes the circle diagram of the stator current. If the stator voltage V1 and the mains frequency f as well as the resistances and reactances remain constant, the stator current depends only on the slip s.

Figure 5.29 shows the curve of the stator current I1 as a function of the slip. The stator voltage V1 is the reference value on the real axis. The current moves on a circle depending on the operating conditions of the machine. This circle is named after Heyland and Ossanna. The stator resistance R1 is neglected for the Heyland circle. This circle clearly shows that the current at the stationary condition is much higher than that near zero-load operation.

With the major simplification Xh ^ oc, the current through Xh becomes zero. Figure 5.30 shows the simplified equivalent circuit with R1 ~ 0 and Xa = Xlo + X'2o. This equivalent circuit is used for the derivation of the equation of Kloss in the section headed 'Speed-torque characteristics and typical generator data' on p223.

The simplified equivalent circuit provides a simplified equation of the current:

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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