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      (1.18)

      where xlax, xlay, and xlaz indicate leakage reactance between phase a (of winding set abc) with each phase of other winding set xyz.

      Phenomenon of mutual coupling is due to the sharing of same slot by stator winding conductor of different phase. This is signified by the term of common mutual leakage reactance (xlm). Value of xlm is dependent on displacement angle (α) and winding pitch, resulting in the variation in harmonic coupling of windings. By neglecting xlm, some variation is noted in the voltage harmonic distortion [13] with no change in transient effect. A detailed procedure for the determination of slot reactance is available [24] together with the standard method to evaluate the machine parameters [25, 26].

Detailed mathematical simulation of six-phase synchronous machine is based on determination of voltage and torque equations in integral form with flux linkage per second and speed as state variable, winding currents as output, connected grid voltage and prime mover torque as input variables. The voltage equations (1.1) to (1.7) together with the flux linkages equations (1.8) to (1.14) are firstly solved for the currents, which are then substituted in the voltage equations. Integral forms of mathematical equations are as follows:

Figure 1.1 Equivalent circuit representation of a six-phase synchronous machine.

      (1.19)

      (1.20)

      (1.21)

      (1.22)

      (1.23)

      (1.24)

      (1.25)

      Current in terms of flux is written as follows:

(1.26)

      (1.27)

      (1.28)

      (1.29)

      (1.30)

      (1.31)

(1.32)

      ψ_mq and ψ_md are defined using state variables as follows:

      (1.33)

      (1.34)

      where

      (1.35)

      (1.36)

      (1.37)

      Developed electromagnetic torque (Te) and rotor dynamic equations for the machine can be expressed as follows:

      (1.38)

      where

      (1.39)

      the developed torque associated with the first winding set abc, and

(1.40)

      the developed torque associated with the second winding set xyz.

(1.41)

(1.42)

      where Te is overall developed electromagnetic torque and Tl is the prime mover torque. In the mathematical modeling, both motoring and generating mode is possible. In this chapter, generating mode is considered, where mechanical input torque Tl is fed from prime mover to generate the electrical output power, resulting the flow of current from machine to connected utility grid.

      Input voltages vabcs and vxyzs in stationary frame coordinate are transformed directly to in rotor reference frame by using the transformation matrix [20, 22] associated with each winding sets abc and xyz, respectively. The voltages applied to the damper windings are not shown, because voltages are zero due to short-circuited windings.

1.3 Linearization of Machine Equations for Stability Analysis

In above section, current and flux linkage per second are both related to each other [Equation (1.26) to (1.32)], one variable vector (current or flux linkage per second) can be taken as state variable. The choice of state variable is generally determined by the application [15].