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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]. Here, current is selected as state variable (i.e., independent variable). Hence, the voltage-current relation of machine in matrix form is expressed as follows:

(1.43)

where

[v] = [vq₁, vd₁, vq₂, vd₂, vKq, vfr, vKd]^T

[i] = [iq₁, id₁, iq₂, id₂, iKq, ifr, iKd]^T

[z] is the impedance matrix defined in the Appendix.

Using the concept of Taylor series expansion, the linearization of machine equations (1.40), (1.41), (1.42), and (1.43) results in a set of equations, which are expressed in matrix form:

(1.44)

where matrix elements are explained in the Appendix.

Grid voltage with constant magnitude and frequency is presented in synchronous reference frame. Hence, it is advantageous to relate the variables in synchronously rotating reference frame (i.e., and to rotor reference frame (i.e., and is given by Equation (1.45).

(1.45)

whereas β shows the phase difference between the terminal voltage of phases a and x. Numerically, the numerical value of both α and β is 30° electrical.

The linearized version of above of nonlinear differential equation (1.45) with suitable approximation (cosΔδ_k = 1 and sinΔδ_k = Δδ_k) results in

(1.46)

Inverse transformation of above equation yields

(1.47)

where Fr and Fe represent the d-q performances indices under steady state.

(1.48)

(1.49)

Substituting Equations (1.46) and (1.47) into Equation (1.44) results in

(1.50)

and is rearranged as

(1.51)

Simplified version of above equation can be written as

(1.52)

In above expressions, the additional subscript “0” represent the value during steady state. Rewriting Equation (1.52) in fundamental form

(1.53)

where

In above linearized model of machine, the effect of mutual coupling between stator winding sets abc and xyz is considered (by using mutual leakage reactance, xlm and xldq). Results are presented in consideration of the asymmetrical six-phase synchronous machine (α = 30° electrical) in comparison with its three-phase equivalent.