**8. Conclusions**

<sup>2</sup> V -(1 - ) I Z abG1 G1 G1 <sup>=</sup> g

V (1 - ) I Z abG2 G2 G2 = g

2 1 2 1

2

æ ö æ ö - =ç - ÷ + - ç ÷ ç ÷

(1 ) *G G abL abL <sup>Z</sup> Z Z IZ Z I Z I*

> 1 1 (1 ) *<sup>a</sup> abL I I* = g

2 2 2 (1 ) *<sup>a</sup> abL I I* = g

Since excitation is assumed to occur, by substituting Eqs. (87), (88), (89), and (90) into Eqs.

1 2 1 2 2 1

*Z Z ZZ*

The characteristics equations derived in the previous sections can be represented by one single

0 0 00 3 3 0 *G G*

01 02 1 2

*Z Z ZZ Z Z ZZ ZZ Z Z*

<sup>æ</sup> ö æ <sup>ö</sup> æ öæ ö <sup>ç</sup> + - - - -ç - ÷ç - ÷= ÷ ç <sup>÷</sup> <sup>ç</sup> ÷ ç <sup>÷</sup> ç ÷ç ÷ <sup>è</sup> ø è <sup>ø</sup> è øè ø

æ ö æ ö -- = - ç ÷ +ç - ÷ ç ÷ è ø è ø

(1 ) *G G abL abL Z Z <sup>Z</sup> IZ Z I Z I*

11 0 1 2 2 0 0

*Z Z*

2 1 2 22 1 1 0 2 0 0

*Z Z*

è ø è ø

Since the line voltages at the generator and the load side are equal, hence,

g

156 New Developments in Renewable Energy

g

(85) and (86), and rearranging yield,

**7. Generalization of steady state model**

general equation of the following form:

It is known that,

(83)

(84)

(85)

(86)

(91)

2

*G a* 1 1 *I I* = (87)

*G a* 2 2 *I I* = (88)

(89)

(90)

2 2

A mathematical model based on the sequence equivalent circuits of the SEIG and the sequence components of the three-phase load was developed to study the performance of the SEIG in the steady state condition. Core loss resistance is included in the model as a function of *Xm*.

Furthermore, the magnetizing reactance *Xm* is taken as a variable in the negative sequence equivalent circuit. The performance of the SEIG was determined for no-load, balanced and unbalanced load and/or excitation for different SEIG and load connections. The operating conditions were found by solving the proposed model iteratively using MATHCAD® software as described in section 1. Self-excited induction generator uses excitation capacitors at the terminals. For a given speed, and load situation, there is a specific value of the excitation capacitor that ensures voltage build up. Residual magnetism is a must in SEIG to initiate excitation. In balanced mode of operation, per phase equivalent circuit is solved to find *F* and *Xm*. Unbalanced operation of SEIG can be analyzed through the method of symmetrical components and the phase voltages may differ from each other notably.

An experimental setup has been built to verify the results obtained from the theoretical model. It is found that the theoretical results are in good agreement with those recorded experimen‐ tally. The model is generalized to cover all connection types of SEIG and/or load. The charac‐ teristic equation of each type may be found by substituting the appropriate parameters, i.e., *β*<sup>1</sup> up to *β*10 from table 1, in the general model.

$$\begin{aligned} & \left( \beta\_1 Z\_0 Z\_p + \beta\_2 Z\_0^2 + \beta\_3 Z\_1 Z\_2 \right) (\beta\_4 Z\_0 Z\_n + \beta\_5 Z\_0^2 + \beta\_6 Z\_1 Z\_2) \\ & - (\beta\_7 Z\_0 Z\_1 + \beta\_8 Z\_2^2) (\beta\_9 Z\_0 Z\_2 + \beta\_{10} Z\_1^2) = 0 \end{aligned}$$
